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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR3" target="_self">is_T3</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V5">Basis_sigma_discrete</a> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="nagata_1.html#V5">Basis_sigma_discrete</a>
 ;<br/><a NAME="E4:28_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="nagata_1.html#V2">sigma_discrete</a> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:28_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nagata_1.html#D5">NAGATA_1:def 5</a></i>;<br/><a NAME="E5:28_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="nagata_1.html#V3">sigma_locally_finite</a>
 <b>by </b><i><a class="ref" href="nagata_1.html#T12">NAGATA_1:12</a></i>;<br/><a NAME="E6:28_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="nagata_1.html#V6">Basis_sigma_locally_finite</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:28_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nagata_1.html#D6">NAGATA_1:def 6</a></i>;<br/><b>hence </b><a NAME="E7:28_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V3">metrizable</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:28_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nagata_2.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR3" target="_self">is_T3</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V5">Basis_sigma_discrete</a> ) iff <font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V3">metrizable</a> )
 <b>by </b><i><a class="ref" href="nagata_2.html#T21" target="_self">Th21</a>, <a class="ref" href="nagata_1.html#T15">NAGATA_1:15</a></i>;<br/>


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