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<div class="add">

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>thus </b><a NAME="E1:31"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> implies for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">B</font> being   <a href="connsp_2.html#M1">Nbhd</a> of <font color="Olive">x</font>,<font color="Maroon">T</font> st <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:31_1"/><b>then </b>
 ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">B</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="connsp_2.html#T9">CONNSP_2:9</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1"/>
 ex <font color="Olive">B</font> being   <a href="connsp_2.html#M1">Nbhd</a> of <font color="Maroon">x</font>,<font color="Maroon">T</font> st <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:31"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">B</font> being   <a href="connsp_2.html#M1">Nbhd</a> of <font color="Olive">x</font>,<font color="Maroon">T</font> st <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:31"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">B</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">x</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:31_2"/><b>then </b>
 ex <font color="Olive">B</font> being   <a href="connsp_2.html#M1">Nbhd</a> of <font color="Maroon">x</font>,<font color="Maroon">T</font> st <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:31_2"/>
 ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">B</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="connsp_2.html#T9">CONNSP_2:9</a></i>;<br/>


</div>
