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<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st not <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/> ex <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; not <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a><br/>

<a NAME="E2:23_1_1"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  holds <br/>(  not <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> or  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:23_1_1_1"/><b>then </b>
 ex <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:23_1_1_1"/>
(  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_1_1"/>
<font color="Maroon">T</font> is <a href="t_0topsp.html#V2">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D3">TSP_1:def 3</a></i>;<br/>


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