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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:81"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V2">closed</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>

<a NAME="E2:81_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V2">closed</a>
 
;<br/>
<a NAME="E3:81_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D5">RCOMP_1:def 5</a></i>;<br/>
<a NAME="E4:81_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i><a class="ref" href="jordan5a.html#T6">JORDAN5A:6</a></i>;<br/>
<a NAME="E5:81_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="txt" href="topreal6.html#E1:81"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal6.html#E81"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="pre_topc.html#T17">PRE_TOPC:17</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>hence </b><a NAME="E6:81_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a><br/>


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

<b>assume </b><a NAME="E3:81"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E4:81"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E5:81"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E6:81"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E1:81"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal6.html#E81"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="jordan5a.html#T6">JORDAN5A:6</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:81"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D5">RCOMP_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:81"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>


</div>
