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<b>let </b><font color="Maroon" title="c1">X</font> be  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/>

<b>assume </b><a NAME="E1:14_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="compact1.html#V3" title="COMPACT1:attr.3">locally-relatively-compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compact1.html#D5" target="_self" title="COMPACT1:def.5">COMPACT1:def 5</a><br/>

<b>consider </b><font color="Maroon" title="c3">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font><b> such that </b><br/><a NAME="E3:14_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c3">U</font> is <a href="compact1.html#V2" title="COMPACT1:attr.2">relatively-compact</a>
 <b>by </b><i><a class="txt" href="compact1.html#E1:14_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="compact1.html#D4" target="_self" title="COMPACT1:def.4">Def8</a></i>;<br/>
<a NAME="E4:14_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">U</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="compact1.html#E3:14_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="compact1.html#D2" target="_self" title="COMPACT1:def.2">Def6</a></i>;<br/>
<a NAME="E5:14_1_1"/>
<font color="Maroon" title="c3">U</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="c3">U</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="E6:14_1_1"/><b>then </b><i><font color="Green" title="E10">A4</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">U</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></i>;<br/>
<a NAME="E7:14_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">U</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E8:14_1_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="compact1.html#E6:14_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:14_1_1"/>
 ex <font color="Olive" title="b1">U</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <br/>( <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b1">U</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )
 <b>by </b><i><a class="txt" href="compact1.html#E4:14_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>


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