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<b>let </b><font color="Maroon" title="c5">X</font>, <font color="Maroon" title="c6">Y</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="c7">f</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">Y</font>;<br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c7">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ;<br/><b>thus </b><a NAME="E2:15_1"/>
<font color="Maroon" title="c7">f</font> is <a href="jordan24.html#V2" title="JORDAN24:attr.2">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan24.html#D4" target="_self" title="JORDAN24:def.4">JORDAN24:def 4</a><br/>

<b>thus </b><a NAME="E1:15_1_1"/>
( <font color="Maroon" title="c8">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="jordan24.html#E1:15_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="tops_2.html#T72" title="TOPS_2:th.72">TOPS_2:72</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:15"/><i><font color="Green" title="E11">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">X</font>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan24.html#D4" target="_self" title="JORDAN24:def.4">JORDAN24:def 4</a><br/>

<a NAME="E4:15"/><i><font color="Green" title="E12">A4</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">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="c5">X</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">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="c6">Y</font> )
 
<b>by </b><i><a class="txt" href="jordan24.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:15"/><i><font color="Green" title="E13">A5</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">X</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="c5">X</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">Y</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="c6">Y</font> )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">X</font>;<br/><b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/><a NAME="E2:15_2"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/><b>hence </b><a NAME="E3:15_2"/>
<font color="Maroon" title="c8">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="jordan24.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="jordan24.html#E4:15"><i><font color="Green" title="E12">A4</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="E7:15"/>
<font color="Maroon" title="c7">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="jordan24.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="jordan24.html#E3:15"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="jordan24.html#E4:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="jordan24.html#E5:15"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="tops_2.html#T72" title="TOPS_2:th.72">TOPS_2:72</a></i>;<br/>


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