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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="E1:47"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 ;<br/>

<a NAME="E2:47"/><b>then </b><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T39" target="_self" title="TOPS_3:th.39">Th39</a></i>;<br/>
<b>assume </b><a NAME="E3:47"/>
<font color="Maroon" title="c3">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 ;<br/>

<a NAME="E4:47"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T18" target="_self" title="TOPS_3:th.18">Th18</a></i>;<br/>
<a NAME="E5:47"/><i><font color="Green" title="E32">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></i>;<br/>
<a NAME="E6:47"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </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#E2:47"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="tops_3.html#E4:47"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="tops_3.html#T30" target="_self" title="TOPS_3:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E7:47"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E5:47"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="tops_3.html#T18" target="_self" title="TOPS_3:th.18">Th18</a></i>;<br/>


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