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<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>given </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font><b> such that </b><a NAME="E1:19"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a><br/>

<b>assume </b><a NAME="E2:19"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="connsp_2.html#V1">locally_connected</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:19_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:19_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E3:19_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:19"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E4:19_1"/><b>then </b><i><font color="Green">E27</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T174">RELAT_1:174</a></i>;<br/><a NAME="E5:19_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:19"><i><font color="Green">E23</font></i></a>, <a class="txt" href="topalg_5.html#E1:19_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="topgrp_1.html#T26">TOPGRP_1:26</a></i>;<br/><a NAME="E6:19_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>10</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:19"><i><font color="Green">E23</font></i></a>, <a class="txt" href="topalg_5.html#E2:19_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="topalg_5.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E7:19_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:19"><i><font color="Green">E24</font></i></a>, <a class="txt" href="topalg_5.html#E4:19_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="topalg_5.html#E5:19_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="connsp_2.html#T24">CONNSP_2:24</a></i>;<br/><b>hence </b><a NAME="E8:19_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:19"><i><font color="Green">E23</font></i></a>, <a class="ref" href="topgrp_1.html#T26">TOPGRP_1:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="connsp_2.html#V1">locally_connected</a>
 <b>by </b><i><a class="ref" href="connsp_2.html#T24">CONNSP_2:24</a></i>;<br/>


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