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<b>let </b><font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">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>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 ;<br/>

<a NAME="E5:31"/><i><font color="Green" title="E15">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green" title="E16">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green" title="E17">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="txt" href="topreala.html#E2:31"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green" title="E18">A</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E9:31"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font> holds <br/> ( <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> iff <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font>;<br/>

<b>thus </b><a NAME="E1:31_1"/>
( <font color="Maroon" title="c6">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="topreala.html#E4:31"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a></i>;<br/>

<b>assume </b><a NAME="E2:31_1"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E3:31_1"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">P</font></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="topreala.html#E8:31"><i><font color="Green" title="E18">A</font></i></a>, <a class="txt" href="topreala.html#E3:31"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1"/>
<font color="Maroon" title="c6">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topreala.html#E1:31"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="topreala.html#E5:31"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="funct_1.html#T164" title="FUNCT_1:th.164">FUNCT_1:164</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:31"/>
<font color="Maroon" title="c5">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topreala.html#E1:31"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="topreala.html#E6:31"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="topreala.html#E7:31"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</a></i>;<br/>


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