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<b>let </b><font color="Maroon" title="c6">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c7">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c6">T</font>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c6">T</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">A</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">T</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<a NAME="E3:16"/>
<font color="Maroon" title="c8">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D1" title="TSEP_1:def.1">TSEP_1:def 1</a></i>;<br/>
<a NAME="E4:16"/><b>then </b>
<font color="Maroon" title="c8">A</font> is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:16"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="connsp_2.html#T23" title="CONNSP_2:th.23">CONNSP_2:23</a></i>;<br/>
<a NAME="E5:16"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">S</font> is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T11" target="_self" title="TOPALG_5:th.11">Th11</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
<font color="Maroon" title="c7">S</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T13" target="_self" title="TOPALG_5:th.13">Th13</a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T12" target="_self" title="TOPALG_5:th.12">Th12</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T13" target="_self" title="TOPALG_5:th.13">Th13</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #)</span>)</span> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D6" title="CONNSP_2:def.6">CONNSP_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E10:16"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#T3" title="TSEP_1:th.3">TSEP_1:3</a></i>;<br/>


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