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

<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">SS</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">TS</font>,<font color="Maroon">SS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">TS</font> is <a href="compts_1.html#V2" target="_self">compact</a>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">SS</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E34</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">SS</font> )
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E6:23"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> holds <br/> ( <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:23_1"/>
( <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> implies <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i>, , <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, , </i>;<br/>

<b>assume </b><a NAME="E2:23_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E3:23_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<a NAME="E4:23_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/>
<a NAME="E5:23_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:23_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E7:23_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<b>hence </b><a NAME="E8:23_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="compts_1.html#D3" target="_self">Def3</a>, <a class="ref" href="compts_1.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:23"/>
<font color="Maroon">f</font> is <a href="tops_2.html#V3">being_homeomorphism</a>
 <b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#T72">TOPS_2:72</a></i>;<br/>


</div>
