<?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="waybel_9.html#L1">TopRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:4_1_1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel33.html#V1" target="_self">lim-inf</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<i><font color="Green">E3</font></i>: the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> = 
 <a href="waybel28.html#K4">xi</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="waybel33.html#E1:4_1_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="waybel33.html#D2" target="_self">Def2</a></i>
<br/>.= 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel28.html#D4">WAYBEL28:def 4</a></i>
;<br/>
<a NAME="E3:4_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<a NAME="E4:4_1_1"/><b>then </b><i><font color="Green">E4</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E2:4_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E5:4_1_1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:4_1_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<a NAME="E3:4_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel33.html#E2:4_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="waybel33.html#E1:4_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:4_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel33.html#E2:4_1_1"><i><font color="Green">E3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:4_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E2:4_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:4_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V2">TopSpace-like</a>
 <b>by </b><i><a class="txt" href="waybel33.html#E4:4_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel33.html#E5:4_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>


</div>
