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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a>  &amp;  ex <font color="Olive" title="b1">Bn</font> being   <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">Bn</font> is  <a href="nagata_1.html#V5" title="NAGATA_1:attr.5">Basis_sigma_discrete</a>  ) iff <font color="Maroon" title="c1">T</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a>  )
 <span class="kw">and </span><br/><a NAME="E2:27_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 ex <font color="Olive" title="b1">Bn</font> being   <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">Bn</font> is  <a href="nagata_1.html#V5" title="NAGATA_1:attr.5">Basis_sigma_discrete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c2">Bn</font> being   <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:27_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c2">Bn</font> is  <a href="nagata_1.html#V5" title="NAGATA_1:attr.5">Basis_sigma_discrete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>;<br/><a NAME="E5:27_1"/>
<font color="Maroon" title="c2">Bn</font> is  <a href="nagata_1.html#V2" title="NAGATA_1:attr.2">sigma_discrete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="nagata_1.html#D5" title="NAGATA_1:def.5">NAGATA_1:def 5</a></span>;<br/><a NAME="E6:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c2">Bn</font> is  <a href="nagata_1.html#V3" title="NAGATA_1:attr.3">sigma_locally_finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#T12" title="NAGATA_1:th.12">NAGATA_1:12</a></span>;<br/><a NAME="E7:27_1"/>
 <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="nagata_1.html#D5" title="NAGATA_1:def.5">NAGATA_1:def 5</a></span>;<br/><a NAME="E8:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">Bn</font> is  <a href="nagata_1.html#V6" title="NAGATA_1:attr.6">Basis_sigma_locally_finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="nagata_1.html#D6" title="NAGATA_1:def.6">NAGATA_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E9:27_1"/>
<font color="Maroon" title="c1">T</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="nagata_2.html#T19" target="_self" title="NAGATA_2:th.19">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:27"/>
( ( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a>  &amp;  ex <font color="Olive" title="b1">Bn</font> being   <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b1">Bn</font> is  <a href="nagata_1.html#V5" title="NAGATA_1:attr.5">Basis_sigma_discrete</a>  ) iff <font color="Maroon" title="c1">T</font> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_2.html#T21" target="_self" title="NAGATA_2:th.21">Th21</a>, <a class="ref" href="nagata_1.html#T15" title="NAGATA_1:th.15">NAGATA_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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