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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">T</font> <a href="compts_1.html#NR3" target="_self">is_T3</a>  &amp; <font color="Maroon">T</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  &amp;  ex <font color="Olive">Bn</font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">T</font> st <font color="Olive">Bn</font> is <a href="nagata_1.html#V6" target="_self">Basis_sigma_locally_finite</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">Bn</font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:42"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">Bn</font> is <a href="nagata_1.html#V6" target="_self">Basis_sigma_locally_finite</a>
 ;<br/>
<a NAME="E4:42"/><i><font color="Green">E58</font></i>: 
(  <a href="nagata_1.html#K2" target="_self">Union</a> <font color="Maroon">Bn</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Bn</font> is <a href="nagata_1.html#V3" target="_self">sigma_locally_finite</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:42"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><br/> for <font color="Olive">U</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> holds <br/> ex <font color="Olive">W</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Olive">W</font> &amp;  <a href="prob_1.html#K2">Union</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> (  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Olive">W</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> &amp; <font color="Olive">W</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font> is <a href="pre_topc.html#V3">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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">U</font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span> ;<br/>
<a NAME="E2:42_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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">k</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">bu</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon">bu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>)
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">bu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">O</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:42_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> &amp; <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:42_1_1_1"/>
<font color="Maroon">bu</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i>, </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">k</font>) <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">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">k</font>) <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">BU</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">BU</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i>
<a NAME="E5:42_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">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">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

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

<b>then consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:42_1_2_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2" target="_self">Union</a> <font color="Maroon">Bn</font> )
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:42_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E6:42_1_2_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E7:42_1_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>)
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:42_1_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:42_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">BU</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E6:42_1"/><i><font color="Green">E86</font></i>: 
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">BU</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">bu</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_3"/>
<font color="Maroon">bu</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">BU</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:42_1_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">bu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E4:42_1_3"/>
<font color="Maroon">bu</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:42_1_3"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">bu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:42_1_3"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> &amp; <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:42_1_3"/>
<font color="Maroon">O</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E10:42_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E11:42_1_3"/>
<font color="Maroon">bu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">BUn</font> = <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span> ;<br/><a NAME="E1:42_1_4"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> is <a href="pcomps_1.html#V1">locally_finite</a>
 <b>by </b><i>, </i>;<br/><a NAME="E2:42_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_4_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:42_1_4_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> &amp; <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 ;<br/>
<b>thus </b><a NAME="E4:42_1_4_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">BUn</font> = <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/><a NAME="E4:42_1_4"/>
<font color="Maroon">BUn</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">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">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

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

<b>then consider </b><font color="Maroon">O</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:42_1_4_2"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> &amp; <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 ;<br/>
<b>thus </b><a NAME="E4:42_1_4_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:42_1_4"/><b>then </b>
<font color="Maroon">BUn</font> is <a href="pcomps_1.html#V1">locally_finite</a>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="pcomps_1.html#T12">PCOMPS_1:12</a></i>;<br/><a NAME="E6:42_1_4"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">BUn</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2">clf</a> <font color="Maroon">BUn</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T23">PCOMPS_1:23</a></i>;<br/><a NAME="E7:42_1_4"/><i><font color="Green">E117</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">BUn</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">ClBu</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_4_3"/>
<font color="Maroon">ClBu</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">BUn</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ClB</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_4_3"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">ClBu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ClB</font> &amp; <font color="Maroon">ClB</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2">clf</a> <font color="Maroon">BUn</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ClB</font> = <font color="Maroon">ClB</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:42_1_4_3"/><i><font color="Green">E120</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ClB</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BUn</font> )
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#D3">PCOMPS_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:42_1_4_3"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:42"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:42_1_4_3"/>
<font color="Maroon">ClBu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><i>, <a class="txt" href="nagata_1.html#E1:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:42_1_4"/>
<font color="Maroon">BUn</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">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">B</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

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

<b>then consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:42_1_4_4"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bn</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 ;<br/>
<a NAME="E4:42_1_4_4"/>
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2" target="_self">Union</a> <font color="Maroon">Bn</font> &amp;  <a href="nagata_1.html#K2" target="_self">Union</a> <font color="Maroon">Bn</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, , <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1_4_4"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E9:42_1_4"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">BUn</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/><a NAME="E10:42_1_4"/><b>then </b>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">BUn</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">BUn</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E11:42_1_4"/>
( <font color="Maroon">BU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">BU</font> <a href="urysohn3.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:42_1"/>
 ex <font color="Olive">W</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <font color="Olive">W</font> &amp;  <a href="prob_1.html#K2">Union</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> (  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Olive">W</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> &amp; <font color="Olive">W</font> <a href="urysohn3.html#K4">.</a> <font color="Olive">n</font> is <a href="pre_topc.html#V3">open</a> ) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:42"/>
<font color="Maroon">T</font> <a href="compts_1.html#NR4" target="_self">is_T4</a> 
 <b>by </b><i/>;<br/>


</div>
