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<b>let </b><font color="Maroon">S</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">K</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><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="E2:52_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E3:52_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</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>, </i>;<br/><a NAME="E4:52_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:52_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:52"/><i><font color="Green">E33</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">K</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </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="E1:52_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:52_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><a NAME="E3:52_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</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>hence </b><a NAME="E4:52_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:52"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i/>;<br/>


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