<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="compts_1.html#NV6" title="COMPTS_1:NV.6">Hausdorff</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="compact1.html#V5" title="COMPACT1:attr.5">locally-compact</a> implies <font color="Maroon" title="c1">X</font> is <a href="compact1.html#V3" title="COMPACT1:attr.3">locally-relatively-compact</a> )</span><br/>

<b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="compact1.html#V5" title="COMPACT1:attr.5">locally-compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="compact1.html#V3" title="COMPACT1:attr.3">locally-relatively-compact</a></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compact1.html#D4" target="_self" title="COMPACT1:def.4">COMPACT1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">U</font> is <a href="compact1.html#V2" title="COMPACT1:attr.2">relatively-compact</a></span><br/>

<b>consider </b><font color="Maroon" title="c3">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font><b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c3">U</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="compact1.html#E1:17_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="compact1.html#D6" target="_self" title="COMPACT1:def.6">Def6</a></i>;<br/>
<a NAME="E4:17_1_1"/>
<font color="Maroon" title="c3">U</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="compact1.html#E3:17_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="compts_1.html#T16" title="COMPTS_1:th.16">COMPTS_1:16</a></i>;<br/>
<a NAME="E5:17_1_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E6:17_1_1"/><b>then </b>
<font color="Maroon" title="c3">U</font> is <a href="compact1.html#V2" title="COMPACT1:attr.2">relatively-compact</a>
 
<b>by </b><i><a class="txt" href="compact1.html#E3:17_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="compact1.html#D2" target="_self" title="COMPACT1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E7:17_1_1"/>
 ex <font color="Olive" title="b1">U</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> st <font color="Olive" title="b1">U</font> is <a href="compact1.html#V2" title="COMPACT1:attr.2">relatively-compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
