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<b>let </b><font color="Maroon">Y</font> be  non <a href="realset2.html#V3">trivial</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>

<b>given </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font><b> such that </b><a NAME="E1:34"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> holds <br/><font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a><br/>

<a NAME="E3:34"/>
(  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 
<b>by </b><i>, <a class="ref" href="tsp_1.html#D3">TSP_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:34"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:34"/><i><font color="Green">E60</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) or ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) )
 ;<br/>
<a NAME="E7:34"/><i><font color="Green">E62</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#D2">WAYBEL25:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span><b> such that </b><br/><a NAME="E9:34"/><i><font color="Green">E64</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> ) or ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i>, , <a class="ref" href="tops_3.html#T76">TOPS_3:76</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> =  <a href="funct_3.html#K5">chi</a> <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<a href="waybel18.html#K9">Sierpinski_Space</a>  <b>by </b><i>, <a class="ref" href="yellow16.html#T48">YELLOW16:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>;<br/><a NAME="E1:34_1"/>
<font color="Maroon">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="tops_3.html#T76">TOPS_3:76</a></i>;<br/><b>hence </b><a NAME="E2:34_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E13:34"/><b>then </b>
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> ,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow16.html#T49">YELLOW16:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> ,<font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/>
<a NAME="E15:34"/>
<font color="Maroon">c</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">i</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <a href="waybel18.html#K9">Sierpinski_Space</a> 
 
<b>by </b><i>, , <a class="ref" href="yellow16.html#T50">YELLOW16:50</a></i>;<br/>
<b>hence </b><a NAME="E16:34"/>
 <a href="waybel18.html#K9">Sierpinski_Space</a>  <a href="waybel18.html#R1">is_Retract_of</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel25.html#D1">WAYBEL25:def 1</a></i>;<br/>


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