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<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">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:30"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 ;<br/>

<a NAME="E2:30"/><b>then </b><i><font color="Green" title="E19">A1</font></i>: 
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#D5" title="TOPS_1:def.5">TOPS_1:def 5</a></i>;<br/>
<a NAME="E3:30"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E4:30"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<b>assume </b><a NAME="E5:30"/>
<font color="Maroon" title="c3">B</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 ;<br/>

<a NAME="E6:30"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E2:30"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="tops_1.html#T85" title="TOPS_1:th.85">TOPS_1:85</a></i>;<br/>
<b>hence </b><a NAME="E7:30"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E4:30"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="tops_3.html#T11" target="_self" title="TOPS_3:th.11">Th11</a></i>;<br/>


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