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<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>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>thus </b><a NAME="E1:82"/>
<font color="Maroon">A</font> <a href="topgen_4.html#K4" target="_self">^0</a>  <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="topgen_4.html#K4" target="_self">^0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</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="E2:82"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="topgen_4.html#K4" target="_self">^0</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:82"/>
for <font color="Olive">N</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>  holds not <font color="Olive">N</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> is <a href="card_4.html#V1">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">N</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>;<br/>

<b>consider </b><font color="Maroon">N1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:82_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">N1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<a NAME="E3:82_1"/>
<font color="Maroon">A</font> <a href="topgen_4.html#K4" target="_self">^0</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">N1</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#T54">PRE_TOPC:54</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:82_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="topgen_4.html#K4" target="_self">^0</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E7:82_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">y'</font> <a href="topgen_4.html#R1" target="_self">is_a_condensation_point_of</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">N1</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y'</font> <b>by </b><i>, , <a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<a NAME="E9:82_1"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i>, <a class="ref" href="topgen_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E10:82_1"/>
<font color="Maroon">N1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<b>hence </b><a NAME="E11:82_1"/>
not <font color="Maroon">N</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i>, <a class="ref" href="card_4.html#T46">CARD_4:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:82"/><b>then </b>
<font color="Maroon">p</font> <a href="topgen_4.html#R1" target="_self">is_a_condensation_point_of</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:82"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="topgen_4.html#K4" target="_self">^0</a> 
 <b>by </b><i/>;<br/>


</div>
