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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V3">almost_discrete</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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="E1:108"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<a NAME="E2:108"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T23">TDLAT_3:23</a>, <a class="ref" href="tdlat_3.html#T24">TDLAT_3:24</a></i>;<br/>
<a NAME="E3:108"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>assume </b><a NAME="E4:108"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 ;<br/>

<a NAME="E5:108"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="txt" href="tex_2.html#E2:108"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tex_2.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E6:108"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tex_2.html#E3:108"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tops_3.html#D2">TOPS_3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:108"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V1" target="_self">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T5" target="_self">Th5</a></i>;<br/>


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