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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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 <font color="Olive" title="b1">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  holds <br/> 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>  holds  not <font color="Olive" title="b2">x</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  implies  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>  holds  not <font color="Olive" title="b1">x</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> )</span><br/>



<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  holds  not <font color="Olive" title="b1">x</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></span><br/>

<span class="kw">given </span><font color="Maroon" title="c3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><a NAME="E3:84"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c3">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">N</font> =  the    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font>;<br/>
<a NAME="E4:84"/>
 not  the    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:84"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="card_3.html#T94" title="CARD_3:th.94">CARD_3:94</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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