<?xml version="1.0"?>
<div><span class="kw">theorem </span><a NAME="T58"><span class="comment"><font color="firebrick">:: TOPS_2:58</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">T</font> being    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> <br/>  for <font color="Olive" title="b2">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> <br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">T</font>,<font color="Olive" title="b2">S</font> holds <br/> ( <font color="Olive" title="b3">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  iff (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">T</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive" title="b4">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">T</font> holds <br/> ( <font color="Olive" title="b4">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  iff <font color="Olive" title="b3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b4">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) ) ) )</div></div>
