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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>;<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>;<br/>



<b>thus </b><a NAME="E1:82"/>
<font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c3">x</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="E2:82"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c3">x</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"><font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E4:82"/>
for <font color="Olive" title="b1">N</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">p</font>  holds not <font color="Olive" title="b1">N</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">N</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">p</font>;<br/>

<b>consider </b><font color="Maroon" title="c6">N1</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="E2:82_1"/><i><font color="Green" title="E45">A2</font></i>: 
( <font color="Maroon" title="c6">N1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N1</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T8" title="CONNSP_2:th.8">CONNSP_2:8</a></i>;<br/>
<a NAME="E3:82_1"/>
<font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">N1</font>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E2:82"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="topgen_4.html#E2:82_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="pre_topc.html#T54" title="PRE_TOPC:th.54">PRE_TOPC:54</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:82_1"/><i><font color="Green" title="E46">A3</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c7">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="topgen_4.html#E5:82_1"><i><font color="Green" title="E46">A3</font></i></a></i>;<br/>
<a NAME="E7:82_1"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c8">y'</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E5:82_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="topgen_4.html#D10" target="_self" title="TOPGEN_4:def.10">Def10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">N1</font> = <font color="Maroon" title="c6">N1</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">y'</font> <b>by </b><i><a class="txt" href="topgen_4.html#E5:82_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="topgen_4.html#E2:82_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="connsp_2.html#T8" title="CONNSP_2:th.8">CONNSP_2:8</a></i>;<br/>
<a NAME="E9:82_1"/><i><font color="Green" title="E48">A5</font></i>: 
not <font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a>, <a class="txt" href="topgen_4.html#E7:82_1"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/>
<a NAME="E10:82_1"/>
<font color="Maroon" title="c9">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E2:82_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<b>hence </b><a NAME="E11:82_1"/>
not <font color="Maroon" title="c5">N</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E9:82_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="card_4.html#T46" title="CARD_4:th.46">CARD_4:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:82"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E6:82"/>
<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 href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> 
 <b>by </b><i><a class="ref" href="topgen_4.html#D10" target="_self" title="TOPGEN_4:def.10">Def10</a></i>;<br/>


</div>
