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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<a NAME="E1:38"/>
( <font color="Maroon" title="c1">X</font> is  <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> implies <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> )
 
;<br/>
<b>hence </b><a NAME="E2:38"/>
( <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> implies ( <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) ) )
 <b>by </b><i><a class="ref" href="tdlat_3.html#T18" target="_self" title="TDLAT_3:th.18">Th18</a></i>;<br/>

<b>assume </b><a NAME="E3:38"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>
 ;<br/>

<b>assume </b><a NAME="E4:38"/><i><font color="Green" title="E22">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<a NAME="E5:38"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E4:38"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tdlat_3.html#E3:38"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="tdlat_3.html#T24" target="_self" title="TDLAT_3:th.24">Th24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:38"/>
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T19" target="_self" title="TDLAT_3:th.19">Th19</a></i>;<br/>


</div>
