<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E19</font></i>: 
(  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> is <a href="tops_2.html#V2">closed</a> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E4:8"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E2:8_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:8"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>assume </b><a NAME="E2:8_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_2_1"/>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_1"><b>suppose </b></a><a NAME="E1:8_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:8_2_1_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E3:8_2_1_1"/><b>then </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:8_2_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_2"><b>suppose </b></a><a NAME="E1:8_2_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:8_2_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_2_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:8_2_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:8_2_1_2_1"/><b>then </b>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:8_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">CG</font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/><a NAME="E4:8_2_1_2"/>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G</font> is <a href="tops_2.html#V2">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</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#D2">TOPS_2:def 2</a><br/>

<b>assume </b><a NAME="E1:8_2_1_2_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:8_2_1_2_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E3:8_2_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:8_2_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
;<br/>
<b>hence </b><a NAME="E5:8_2_1_2_2"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:8_2_1_2"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">CG</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/><a NAME="E6:8_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G'</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T11">TOPS_2:11</a></i>;<br/><a NAME="E7:8_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E8:8_2_1_2"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:8_2_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:8"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E2:8_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font>, <font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:8_3"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:8_3"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:8_3"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:8_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:8_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:8_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>;<br/>
<a NAME="E10:8_3"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:8_3"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:8"/>
<font color="Maroon">T</font> is   <a href="pre_topc.html#NM1">TopSpace</a>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>


</div>
