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

<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/>

<b>assume </b><a NAME="E1:71"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="yellow_8.html#V3">sober</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V6">locally-compact</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel12.html#D6" target="_self">WAYBEL12:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:71"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>and </b><br/><a NAME="E3:71"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V1">dense</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E5:71"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E6:71"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="waybel12.html#V1" target="_self">dense</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:71_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></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="E1:71_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:71_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:71"><i><font color="Green">E45</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:71_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:71"><i><font color="Green">E45</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel12.html#D2" target="_self">WAYBEL12:def 2</a><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_1"><b>suppose </b></a><a NAME="E1:71_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:71_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:71_2_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="yellow_8.html#V2">irreducible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71_2_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:71"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel12.html#E2:71"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel12.html#E6:71"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel12.html#E1:71_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel12.html#T47" target="_self">Th47</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:71_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:71_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:71"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_8.html#D6">YELLOW_8:def 6</a></i>;<br/>
<a NAME="E7:71_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#D4">YELLOW_8:def 4</a></i>;<br/>
<a NAME="E8:71_2_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel12.html#E5:71_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="yellow_8.html#T25">YELLOW_8:25</a></i>;<br/>
<a NAME="E9:71_2_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:71_2_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:71_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:71_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel12.html#E1:71_2_1_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E4:71_2_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:71_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:71"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel12.html#E1:71_2_1_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E2:71_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:71_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/><a NAME="E3:71_2_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></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_1.html#T30">TOPS_1:30</a></i>;<br/><b>assume </b><a NAME="E4:71_2_1_1_1_1"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E5:71_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/><a NAME="E6:71_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E7:71_2_1_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E8:71_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="waybel12.html#E8:71_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel12.html#E3:71_2_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E9:71_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E10:71_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E11:71_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel12.html#E4:71_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:71_2_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:71_2_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<a NAME="E11:71_2_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:71_2_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:71_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E14:71_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E9:71_2_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="waybel12.html#E13:71_2_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E15:71_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="waybel12.html#E11:71_2_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:71_2_1_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:71_2_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T80">TOPS_1:80</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_2"><b>suppose </b></a><a NAME="E1:71_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then </b> <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>

;<br/><b>hence </b><a NAME="E3:71_2_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T16">TOPS_3:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
