<?xml version="1.0"?>
<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>set </b><font color="Maroon">F</font> =  <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font>;<br/>
<a NAME="E1:16"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/><span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:16_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>set </b><font color="Maroon">A'</font> = <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="t_0topsp.html#K2" target="_self">Indiscernible</a> <font color="Maroon">T</font>;<br/>
<a NAME="E2:16_1"/><i><font color="Green">E19</font></i>: 
 <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font> <a href="funct_2.html#R1">=</a>  <a href="borsuk_1.html#K2">proj</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K2" target="_self">Indiscernible</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D11">BORSUK_1:def 11</a></i>;<br/>
<a NAME="E3:16_1"/>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K2" target="_self">Indiscernible</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E1:16_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="borsuk_1.html#T31">BORSUK_1:31</a></i>;<br/>
<a NAME="E5:16_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="t_0topsp.html#K2" target="_self">Indiscernible</a> <font color="Maroon">T</font> &amp; <font color="Olive">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> holds <br/><font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:16_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E1:16_1"><i><font color="Green">E16</font></i></a>, , <a class="ref" href="borsuk_1.html#T33">BORSUK_1:33</a></i>;<br/>
<a NAME="E7:16_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:16_1"/>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16"/>
 <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font> is <a href="t_0topsp.html#V1" target="_self">open</a>
 <b>by </b><i/>;<br/>


</div>
