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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <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>assume </b><a NAME="E1:93"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:93"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E3:93"/><i><font color="Green">E57</font></i>: 
 <a href="tops_1.html#K2" target="_self">Fr</a> <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>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:93"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E2:93"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tops_1.html#T73" target="_self">Th73</a></i>;<br/>
<a NAME="E5:93"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:93"/>
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="tops_1.html#E2:93"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tops_1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E7:93"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="tops_1.html#E5:93"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tops_1.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E8:93"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E9:93"/><i><font color="Green">E60</font></i>: 
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tops_1.html#E3:93"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tops_1.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E10:93"/>
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" target="_self">Th44</a></i>;<br/>
<a NAME="E11:93"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E9:93"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:93"/><i><font color="Green">E62</font></i>: 
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:93_1"/>
not  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 ;<br/>

<a NAME="E2:93_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tops_1.html#T84" target="_self">Th84</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E4:93_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tops_1.html#E2:93_1"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E5:93_1"/><b>then </b><i><font color="Green">E64</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E6:93_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E11:93"><i><font color="Green">E61</font></i></a>, <a class="txt" href="tops_1.html#E2:93_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:93_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tops_1.html#E8:93"><i><font color="Green">E59</font></i></a>, <a class="txt" href="tops_1.html#E2:93_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="tops_1.html#E5:93_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tops_1.html#T39" target="_self">Th39</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E13:93"/>
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V3" target="_self">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tops_1.html#E12:93"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tops_1.html#T93" target="_self">Th93</a></i>;<br/>


</div>
