<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR3" title="COMPTS_1:NR.3">is_T3</a>  &amp; <font color="Maroon" title="c1">T</font> <a href="urysohn1.html#NR1" title="URYSOHN1:NR.1">is_T1</a>  &amp;  ex <font color="Olive" title="b1">Bn</font> being  <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">Bn</font> is <a href="nagata_1.html#V6" title="NAGATA_1:attr.6">Basis_sigma_locally_finite</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">Bn</font> being   <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:42"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">Bn</font> is <a href="nagata_1.html#V6" title="NAGATA_1:attr.6">Basis_sigma_locally_finite</a>
 ;<br/>
<a NAME="E4:42"/><i><font color="Green" title="E23">A3</font></i>: 
(  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c2">Bn</font> is <a href="nagata_1.html#V3" title="NAGATA_1:attr.3">sigma_locally_finite</a> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E3:42"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="nagata_1.html#D6" target="_self" title="NAGATA_1:def.6">Def6</a></i>;<br/>
<a NAME="E5:42"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">U</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b3">W</font> &amp;  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp; ( for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">W</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp; <font color="Olive" title="b3">W</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b4">n</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">U</font> be  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E24">A4</font></i>: 
( <font color="Maroon" title="c3">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c4">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> $1 &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span> ;<br/>
<a NAME="E2:42_1"/><i><font color="Green" title="E25">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">bu</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon" title="c7">bu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">k</font>)
 ;<br/><b>then consider </b><font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c7">bu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:42_1_1_1"/><i><font color="Green" title="E27">A7</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">O</font> &amp; <font color="Maroon" title="c9">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_1_1"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><b>thus </b><a NAME="E6:42_1_1_1"/>
<font color="Maroon" title="c7">bu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="nagata_1.html#E5:42_1_1_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:42_1_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">BU</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green" title="E26">A8</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c5">BU</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="nagata_1.html#E2:42_1"><i><font color="Green" title="E25">A5</font></i></a>);<br/></i>
<a NAME="E5:42_1"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c5">BU</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c5">BU</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:42_1_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:42_1_2_1"/><i><font color="Green" title="E28">A10</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c7">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:42"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="nagata_1.html#E4:42"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="nagata_1.html#E1:42_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="nagata_1.html#T19" target="_self" title="NAGATA_1:th.19">Th19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:42_1_2_1"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c7">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>
<a NAME="E6:42_1_2_1"/>
<font color="Maroon" title="c7">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c8">k</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="nagata_1.html#E5:42_1_2_1"><i><font color="Green" title="E29">A11</font></i></a></i>;<br/>
<a NAME="E7:42_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">k</font>)
 
<b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_2_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:42_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">BU</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E4:42_1"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:42_1_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c5">BU</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c5">BU</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:42_1"/><i><font color="Green" title="E28">A12</font></i>: 
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c5">BU</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">bu</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_3"/>
<font color="Maroon" title="c6">bu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c5">BU</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:42_1_3"/><i><font color="Green" title="E29">A13</font></i>: 
<font color="Maroon" title="c6">bu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">BU</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>
<a NAME="E4:42_1_3"/>
<font color="Maroon" title="c6">bu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c7">k</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="nagata_1.html#E4:42_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="nagata_1.html#E3:42_1_3"><i><font color="Green" title="E29">A13</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:42_1_3"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c6">bu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c7">k</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:42_1_3"/><i><font color="Green" title="E31">A15</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">O</font> &amp; <font color="Maroon" title="c9">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c7">k</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E6:42_1_3"><i><font color="Green" title="E30">A14</font></i></a></i>;<br/>
<a NAME="E9:42_1_3"/>
<font color="Maroon" title="c9">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">O</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E10:42_1_3"/><b>then </b>
<font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E8:42_1_3"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E11:42_1_3"/>
<font color="Maroon" title="c6">bu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">U</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E6:42_1_3"><i><font color="Green" title="E30">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>set </b><font color="Maroon" title="c7">BUn</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span> ;<br/><a NAME="E1:42_1_4"/><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>
 <b>by </b><i><a class="txt" href="nagata_1.html#E4:42"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="nagata_1.html#D3" target="_self" title="NAGATA_1:def.3">Def3</a></i>;<br/><a NAME="E2:42_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_4_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:42_1_4_1"/><i><font color="Green" title="E30">A17</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">O</font> &amp; <font color="Maroon" title="c9">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 ;<br/>
<b>thus </b><a NAME="E4:42_1_4_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_4_1"><i><font color="Green" title="E30">A17</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c8">BUn</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E4:42_1_4"/>
<font color="Maroon" title="c8">BUn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_4_2"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">BUn</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:42_1_4_2"/><i><font color="Green" title="E30">A18</font></i>: 
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">O</font> &amp; <font color="Maroon" title="c10">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c10">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 ;<br/>
<b>thus </b><a NAME="E4:42_1_4_2"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_4_2"><i><font color="Green" title="E30">A18</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:42_1_4"/><b>then </b>
<font color="Maroon" title="c8">BUn</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:42_1_4"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="pcomps_1.html#T12" title="PCOMPS_1:th.12">PCOMPS_1:12</a></i>;<br/><a NAME="E6:42_1_4"/><b>then </b><i><font color="Green" title="E30">A19</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">BUn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c8">BUn</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T23" title="PCOMPS_1:th.23">PCOMPS_1:23</a></i>;<br/><a NAME="E7:42_1_4"/><i><font color="Green" title="E31">A20</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">BUn</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">ClBu</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_4_3"/>
<font color="Maroon" title="c9">ClBu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">BUn</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">ClB</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_4_3"/><i><font color="Green" title="E32">A21</font></i>: 
( <font color="Maroon" title="c9">ClBu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ClB</font> &amp; <font color="Maroon" title="c10">ClB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c8">BUn</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E6:42_1_4"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">ClB</font> = <font color="Maroon" title="c10">ClB</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_4_3"><i><font color="Green" title="E32">A21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:42_1_4_3"/><i><font color="Green" title="E33">A22</font></i>: 
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ClB</font> &amp; <font color="Maroon" title="c12">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">BUn</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_4_3"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:42_1_4_3"/><i><font color="Green" title="E34">A23</font></i>: 
( <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q</font> &amp; <font color="Maroon" title="c13">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c13">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E6:42_1_4_3"><i><font color="Green" title="E33">A22</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:42_1_4_3"/>
<font color="Maroon" title="c9">ClBu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">U</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_4_3"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="nagata_1.html#E6:42_1_4_3"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="nagata_1.html#E8:42_1_4_3"><i><font color="Green" title="E34">A23</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:42_1_4"/>
<font color="Maroon" title="c8">BUn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_4_4"/>
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">BUn</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:42_1_4_4"/><i><font color="Green" title="E32">A24</font></i>: 
( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Q</font> &amp; <font color="Maroon" title="c10">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Bn</font> <a href="nagata_1.html#K1" title="NAGATA_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c10">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 ;<br/>
<a NAME="E4:42_1_4_4"/>
( <font color="Maroon" title="c10">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> &amp;  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E4:42"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="nagata_1.html#E3:42_1_4_4"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="cantor_1.html#D2" title="CANTOR_1:def.2">CANTOR_1:def 2</a>, <a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1_4_4"/>
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:42_1_4_4"><i><font color="Green" title="E32">A24</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:42_1_4"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">BUn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/><a NAME="E10:42_1_4"/><b>then </b>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">BUn</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">BUn</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E7:42_1_4"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E11:42_1_4"/>
( <font color="Maroon" title="c5">BU</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c6">n</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">BU</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E4:42_1"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:42_1"/>
 ex <font color="Olive" title="b1">W</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b1">W</font> &amp;  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Olive" title="b1">W</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b2">n</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) ) )
 <b>by </b><i><a class="txt" href="nagata_1.html#E5:42_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="nagata_1.html#E6:42_1"><i><font color="Green" title="E28">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:42"/>
<font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR4" title="COMPTS_1:NR.4">is_T4</a> 
 <b>by </b><i><a class="ref" href="nagata_1.html#T18" target="_self" title="NAGATA_1:th.18">Th18</a></i>;<br/>


</div>
