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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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>  st ( for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b1">A</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Olive" title="b2">x</font>,<font color="Maroon" title="c1">T</font> ) holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon" title="c2">A</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">T</font> ) implies <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon" title="c2">A</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<a NAME="E2:32"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b2">B</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">T</font> st <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies  ex <font color="Olive" title="b1">B</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E2:32_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1"/>
<font color="Maroon" title="c2">A</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E1:32"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="urysohn1.html#E1:32_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">B</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:32_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 ;<br/>
<b>take </b>
<font color="Maroon" title="c4">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>thus </b><a NAME="E4:32_1_1"/>
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:32_1_1"><i><font color="Green" title="E22">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">Nbhd</a> of <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="urysohn1.html#T24" target="_self" title="URYSOHN1:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
