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<div><span class="kw">theorem </span><a NAME="T19"><span class="comment"><font color="firebrick">:: TOPS_3:19</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a><br/>  for <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">X</font> holds <br/> ( <font color="Olive" title="b2">A</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  iff  for <font color="Olive" title="b3">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b3">C</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  holds <br/><font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">X</font> ) <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T5" onmouseover="rs('tops_1/T5')" onmouseout="rh()">TOPS_1:5</a>, <a class="ref" href="pre_topc.html#T18" onmouseover="rs('pre_topc/T18')" onmouseout="rh()">PRE_TOPC:18</a></span>;<br/></div></div>
