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

<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"> ( <font color="Maroon" title="c1">T</font> is <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a> implies  ex <font color="Olive" title="b1">B</font> being   <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">B</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> )</span><br/>

<b>assume </b><a NAME="E1:89"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>
 ; <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="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">B</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a></span><br/>

<b>consider </b><font color="Maroon" title="c2">B</font> being    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">B1</font> being    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:89"/><i><font color="Green" title="E49">A2</font></i>: 
( <font color="Maroon" title="c3">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="ref" href="topgen_2.html#T20" title="TOPGEN_2:th.20">TOPGEN_2:20</a></i>;<br/>
<a NAME="E5:89"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="topgen_4.html#E1:89"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="topgen_4.html#E4:89"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="waybel23.html#D6" title="WAYBEL23:def.6">WAYBEL23:def 6</a></i>;<br/>
<a NAME="E6:89"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E7:89"/><b>then </b>
<font color="Maroon" title="c3">B1</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T44" title="CARD_4:th.44">CARD_4:44</a>, <a class="ref" href="waybel12.html#T2" title="WAYBEL12:th.2">WAYBEL12:2</a></i>;<br/>
<b>hence </b><a NAME="E8:89"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">B</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
