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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>thus </b><a NAME="E2:53"/>
( <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> implies  ex <font color="Olive" title="b1">X1</font> being non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">X1</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> &amp; <font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X1</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>
 ;<br/>

<a NAME="E2:53_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 
<b>by </b><i><a class="ref" href="tex_3.html#D4" target="_self" title="TEX_3:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:53_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font>
 <b>and </b><br/><a NAME="E5:53_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c4">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E6:53_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c4">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T27" title="TOPS_3:th.27">TOPS_3:27</a></i>;<br/>
<a NAME="E7:53_1"/>
not <font color="Maroon" title="c4">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="tex_3.html#E4:53_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">X1</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E9:53_1"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c5">X1</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a>
 <b>and </b><br/><a NAME="E10:53_1"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c4">C</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">X1</font>
 <b>by </b><i><a class="txt" href="tex_3.html#E5:53_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="tex_3.html#E6:53_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="tex_3.html#T41" target="_self" title="TEX_3:th.41">Th41</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">X1</font>
;<br/>

<b>thus </b><a NAME="E11:53_1"/>
( <font color="Maroon" title="c5">X1</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> &amp; <font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c5">X1</font> )
 <b>by </b><i><a class="txt" href="tex_3.html#E4:53_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="tex_3.html#E9:53_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="tex_3.html#E10:53_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c4">X1</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E3:53"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c4">X1</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a>
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c6">C</font> = <font color="Maroon" title="c5">C</font>;<br/><b>thus </b><a NAME="E1:53_2"/>
( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c6">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="tex_3.html#E3:53"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="tex_3.html#E4:53"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="tex_3.html#D3" target="_self" title="TEX_3:def.3">Def3</a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a>, <a class="ref" href="tsep_1.html#T11" title="TSEP_1:th.11">TSEP_1:11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:53"/><b>then </b>
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="c1">X</font>  st <font color="Olive" title="b1">A</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="c2">X0</font> holds <br/><font color="Olive" title="b1">A</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T27" title="TOPS_3:th.27">TOPS_3:27</a></i>;<br/>
<b>hence </b><a NAME="E8:53"/>
<font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tex_3.html#D4" target="_self" title="TEX_3:def.4">Def4</a></i>;<br/>


</div>
