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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T3</a>  &amp; <font color="Maroon" title="c1">T</font> <a href="urysohn1.html#NR1" title="URYSOHN1:NR.1">is_T1</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"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T3</a>  &amp; <font color="Maroon" title="c1">T</font> <a href="urysohn1.html#NR1" title="URYSOHN1:NR.1">is_T1</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>then consider </b><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><b> such that </b><br/><a NAME="E3:27_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c2">Bn</font> is <a href="nagata_1.html#V5" title="NAGATA_1:attr.5">Basis_sigma_discrete</a>
 ;<br/><a NAME="E4:27_1"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c2">Bn</font> is <a href="nagata_1.html#V2" title="NAGATA_1:attr.2">sigma_discrete</a> &amp;  <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#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:27_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="nagata_1.html#D5" title="NAGATA_1:def.5">NAGATA_1:def 5</a></i>;<br/><a NAME="E5:27_1"/><b>then </b>
<font color="Maroon" title="c2">Bn</font> is <a href="nagata_1.html#V3" title="NAGATA_1:attr.3">sigma_locally_finite</a>
 <b>by </b><i><a class="ref" href="nagata_1.html#T12" title="NAGATA_1:th.12">NAGATA_1:12</a></i>;<br/><a NAME="E6:27_1"/><b>then </b>
<font color="Maroon" title="c2">Bn</font> is <a href="nagata_1.html#V6" title="NAGATA_1:attr.6">Basis_sigma_locally_finite</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:27_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="nagata_1.html#D6" title="NAGATA_1:def.6">NAGATA_1:def 6</a></i>;<br/><b>hence </b><a NAME="E7:27_1"/>
<font color="Maroon" title="c1">T</font> is <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:27_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="nagata_2.html#T19" target="_self" title="NAGATA_2:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27"/>
( ( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T3</a>  &amp; <font color="Maroon" title="c1">T</font> <a href="urysohn1.html#NR1" title="URYSOHN1:NR.1">is_T1</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> )
 <b>by </b><i><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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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