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<b>let </b><font color="Maroon" title="c1">T</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="E1:25"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ;<br/>

<a NAME="E2:25"/>
for <font color="Olive" title="b1">FX</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">FX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b1">FX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> holds <br/> ex <font color="Olive" title="b2">GX</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Olive" title="b2">GX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b2">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">FX</font> &amp; <font color="Olive" title="b2">GX</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">FX</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c2">FX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c2">FX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">GX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green" title="E21">A3</font></i>: 
( <font color="Maroon" title="c3">GX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">FX</font> &amp; <font color="Maroon" title="c3">GX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c3">GX</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="compts_1.html#D3" title="COMPTS_1:def.3">COMPTS_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">GX</font>
;<br/>

<b>thus </b><a NAME="E4:25_1"/>
<font color="Maroon" title="c3">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1"/>
for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">GX</font> holds <br/><font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E1:25_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="pcomps_1.html#E3:25_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:25_1_1"/>
<font color="Maroon" title="c3">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:25_1"/>
<font color="Maroon" title="c3">GX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:25_1"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:25_1"/>
<font color="Maroon" title="c3">GX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">FX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:25_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="setfam_1.html#T17" title="SETFAM_1:th.17">SETFAM_1:17</a></i>;<br/>

<b>thus </b><a NAME="E7:25_1"/>
<font color="Maroon" title="c3">GX</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:25_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="pcomps_1.html#T13" target="_self" title="PCOMPS_1:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
<font color="Maroon" title="c1">T</font> is <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D4" target="_self" title="PCOMPS_1:def.4">Def4</a></i>;<br/>


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