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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/>

<b>set </b><font color="Maroon" title="c2">T</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font>;<br/>
<b>thus </b><a NAME="E1:55"/>
(  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is <a href="compl_sp.html#V8" title="COMPL_SP:attr.8">countably_compact</a> )
 <b>by </b><i><a class="ref" href="compl_sp.html#T20" target="_self" title="COMPL_SP:th.20">Th21</a></i>;<br/>

<b>assume </b><a NAME="E2:55"/><i><font color="Green" title="E49">A1</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is <a href="compl_sp.html#V8" title="COMPL_SP:attr.8">countably_compact</a>
 ;<br/>

<a NAME="E3:55"/><b>then </b>
<font color="Maroon" title="c1">M</font> is <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a>
 
<b>by </b><i><a class="ref" href="compl_sp.html#T32" target="_self" title="COMPL_SP:th.32">Th33</a></i>;<br/>
<a NAME="E4:55"/><b>then </b><i><font color="Green" title="E50">A2</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> is <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a>
 
<b>by </b><i><a class="ref" href="compl_sp.html#T33" target="_self" title="COMPL_SP:th.33">Th34</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D3" title="COMPTS_1:def.3">COMPTS_1:def 3</a><br/>

<b>assume </b><a NAME="E5:55"/><i><font color="Green" title="E51">A3</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c3">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><b> such that </b><br/><a NAME="E7:55"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c4">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c4">G</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c4">G</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E4:55"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="compl_sp.html#E5:55"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="compl_sp.html#T34" target="_self" title="COMPL_SP:th.34">Th35</a></i>;<br/>
<a NAME="E8:55"/>
<font color="Maroon" title="c4">G</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E5:55"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="compl_sp.html#E7:55"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">H</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><b> such that </b><br/><a NAME="E10:55"/><i><font color="Green" title="E53">A5</font></i>: 
( <font color="Maroon" title="c5">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c5">H</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c5">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:55"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="compl_sp.html#E7:55"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="compl_sp.html#D9" target="_self" title="COMPL_SP:def.9">DEF9</a></i>;<br/>
<a NAME="E11:55"/>
<font color="Maroon" title="c5">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E7:55"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="compl_sp.html#E10:55"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E12:55"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E10:55"><i><font color="Green" title="E53">A5</font></i></a></i>;<br/>


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