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<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">K</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<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/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:49_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow_8.html#T18">YELLOW_8:18</a></i>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E3:49_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:49_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:49_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:49_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:49"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 ;<br/>

<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span> ;<br/>
<a NAME="E3:49"/>
<span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:49_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:49_2"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:49_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E6:49"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:49_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:49_3"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
;<br/>
<a NAME="E3:49_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:49_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<a NAME="E7:49"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_4_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:49_4_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:49_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:49_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E2:49_4"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:49_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:49_4_2"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:49_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:49_4"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<b>hence </b><a NAME="E4:49_4"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:49"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>


</div>
