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

<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>;<br/><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>thus </b><a NAME="E1:28"/>
( <font color="Maroon">A</font> is <a href="tops_1.html#V3">nowhere_dense</a> implies for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> &amp; <font color="Olive">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">H</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/>

<a NAME="E2:28_1"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D5">TOPS_1:def 5</a></i>;<br/>
<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E3:28_1"/>
( <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:28_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> )
 <b>and </b><br/><a NAME="E6:28_1"/><i><font color="Green">E24</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T87">TOPS_1:87</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E7:28_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E8:28_1"/>
( <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="tops_3.html#D1" target="_self">Def1</a>, <a class="ref" href="tops_3.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


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

<b>assume </b><a NAME="E2:28"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> &amp; <font color="Olive">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">H</font> )
 ;<br/>

<a NAME="E3:28"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> &amp; <font color="Olive">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:28_2"/>
( <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> )
 <b>and </b><br/><a NAME="E4:28_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E5:28_2"/>
( <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="tops_3.html#D1" target="_self">Def1</a>, <a class="ref" href="tops_3.html#T10" target="_self">Th10</a>, <a class="ref" href="tsep_1.html#T40">TSEP_1:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:28"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T87">TOPS_1:87</a></i>;<br/>
<b>hence </b><a NAME="E5:28"/>
<font color="Maroon">A</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/>


</div>
