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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="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:68"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E2:68"/><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel18.html#D9">WAYBEL18:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> ,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:68"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/>
<a NAME="E5:68"/><i><font color="Green">E46</font></i>: 
( 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; not 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</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="E1:68_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:68_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) or ( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) or ( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="yellow16.html#E1:68"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E3:68_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="yellow16.html#E2:68"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/><a NAME="E4:68_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="yellow16.html#E4:68"><i><font color="Green">E45</font></i></a>, <a class="txt" href="yellow16.html#E5:68"><i><font color="Green">E46</font></i></a>, <a class="txt" href="yellow16.html#E2:68_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><a NAME="E5:68_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> 
 <b>by </b><i><a class="ref" href="waybel18.html#D9">WAYBEL18:def 9</a></i>;<br/><b>hence </b><a NAME="E6:68_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:68"/>
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> ,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


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