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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="urysohn1.html#V1">being_T1</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:61"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="topgen_5.html#V1" target="_self">Tychonoff</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E2:61"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E3:61"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T28">YELLOW_9:28</a></i>;<br/>
<a NAME="E4:61"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E2:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E5:61"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<b>hence </b><a NAME="E6:61"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:61"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topgen_5.html#E2:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="topgen_5.html#D4" target="_self">Def4</a></i>;<br/>


</div>
