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

<b>let </b><font color="Maroon">G</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">G</font>;<br/>



<b>assume </b><a NAME="E1:111"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:111_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E2:111_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/><a NAME="E3:111_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#D4">TOPS_1:def 4</a></i>;<br/><a NAME="E4:111_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="tops_1.html#D3">TOPS_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<br/><a NAME="E6:111_1"/>
<font color="Maroon">A2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:111_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E9:111_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E10:111_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E13:111_1"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A1</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><a NAME="E14:111_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">p</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#T43">CONNSP_1:43</a></i>;<br/><a NAME="E15:111_1"/>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E16:111_1"/><b>then </b>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">B0</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">p</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:111_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">B0</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/><a NAME="E19:111_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T40">CONNSP_1:40</a></i>;<br/><a NAME="E20:111_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E21:111_1"/><b>then </b>
<font color="Maroon">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><b>hence </b><a NAME="E22:111_1"/>
 ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">B</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">B</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T25" target="_self">Th25</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:111"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">B</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">B</font> )
 ;<br/>

<a NAME="E4:111"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:111_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:111_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_2_1_1"><b>suppose </b></a><a NAME="E1:111_2_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:111_2_1_1"/>
for <font color="Olive">G1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>  st <font color="Olive">G1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">G1</font> holds <br/><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:111_2_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:111_2_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:111_2_1_1_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">B</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">G1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan2c.html#T24" target="_self">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon">B1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:111_2_1_1_1"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B1</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E7:111_2_1_1_1"/>
<font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:111_2_1_1_1"/><b>then </b>
<font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E9:111_2_1_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E10:111_2_1_1_1"/>
<font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <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="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E11:111_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E12:111_2_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:111_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T22" target="_self">Th22</a>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_2_1_2"><b>suppose </b></a><a NAME="E1:111_2_1_2"/>
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:111_2_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T22" target="_self">Th22</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:111_2_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E4:111_2_1_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><b>hence </b><a NAME="E5:111_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:111"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
;<br/>
<a NAME="E6:111"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T22" target="_self">Th22</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E7:111"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3">TOPS_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:111"/>
<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#D4">TOPS_1:def 4</a></i>;<br/>


</div>
