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<div class="add">

<b>set </b><font color="Maroon" title="c1">S</font> =  <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ;<br/>
<a NAME="E1:3_1_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  st  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel18.html#D9" title="WAYBEL18:def.9">WAYBEL18:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">p</font> = 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>thus </b><a NAME="E3:3_1_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="waybel25.html#T2" target="_self" title="WAYBEL25:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:3_1_1"/>
not  <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  is <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a>
 <b>by </b><i><a class="ref" href="yellow_8.html#T35" title="YELLOW_8:th.35">YELLOW_8:35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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