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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:27"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V2">boundary</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/>

<b>take </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E2:27_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2">boundary</a> )
 <b>by </b><i><a class="txt" href="tops_3.html#E1:27_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="tops_1.html#D5">TOPS_1:def 5</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:27"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<a NAME="E4:27"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="tops_3.html#E2:27"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/>
<a NAME="E5:27"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2">boundary</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E3:27"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tops_3.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E6:27"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D5">TOPS_1:def 5</a></i>;<br/>


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