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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <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>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a>
 <b>and </b><br/><a NAME="E3:45"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E4:45"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="t_0topsp.html#V1">open</a>
 ;<br/>

<a NAME="E5:45"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:45"/><b>then </b><i><font color="Green">E45</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">S</font>
 
;<br/>
<a NAME="E7:45"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E8:45"/><b>then </b><i><font color="Green">E46</font></i>: 
 <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">T</font>
 
;<br/>
<a NAME="E9:45"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</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#V3">open</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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">S</font>;<br/>

<b>thus </b><a NAME="E1:45_1"/>
( <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</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#V3">open</a> )
 <b>by </b><i>, <a class="ref" href="t_0topsp.html#D2">T_0TOPSP:def 2</a></i>;<br/>

<b>assume </b><a NAME="E2:45_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:45_1"/><b>then </b>
<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#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E4:45_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T164">FUNCT_1:164</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:45"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , <a class="ref" href="topgrp_1.html#T25">TOPGRP_1:25</a></i>;<br/>


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