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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Y</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="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/><b>thus </b><a NAME="E2:15_1"/>
<font color="Maroon">f</font> is <a href="jordan24.html#V2" target="_self">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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan24.html#D4" target="_self">JORDAN24:def 4</a><br/>

<b>thus </b><a NAME="E1:15_1_1"/>
( <font color="Maroon">A</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">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T72">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">E63</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan24.html#D4" target="_self">JORDAN24:def 4</a><br/>

<a NAME="E4:15"/><i><font color="Green">E75</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">X</font> &amp;  <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">Y</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:15"/><i><font color="Green">E66</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E2:15_2"/><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">A</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/><b>hence </b><a NAME="E3:15_2"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</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="E7:15"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , , <a class="ref" href="tops_2.html#T72">TOPS_2:72</a></i>;<br/>


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