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<span class="kw">reconsider </span><font color="Maroon" title="c1">R</font> =  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T12" onmouseover="rs('numbers/T12')" onmouseout="rh()">NUMBERS:12</a>, <a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<a NAME="E2:48"/>
(  <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 1</span>)</span> &amp;  <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" onmouseover="rs('euclid/D8')" onmouseout="rh()">EUCLID:def 8</a></span>;<br/>
<a NAME="E3:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A1</font></span>: 
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #),<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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #)</span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#T4" target="_self" onmouseover="rs('metrizts/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="topreal7.html#T25" onmouseover="rs('topreal7/T25')" onmouseout="rh()">TOPREAL7:25</a></span>;<br/>
<a NAME="E4:48"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T15" onmouseover="rs('borsuk_5/T15')" onmouseout="rh()">BORSUK_5:15</a></span>;<br/>
<a NAME="E5:48"/><span class="kw">then </span>
<font color="Maroon" title="c1">R</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#D3" onmouseover="rs('tops_1/D3')" onmouseout="rh()">TOPS_1:def 3</a></span>;<br/>
<a NAME="E6:48"/><span class="kw">then </span>
 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is  <a href="topgen_1.html#V7" title="TOPGEN_1:attr.7">separable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#T6" onmouseover="rs('topgen_4/T6')" onmouseout="rh()">TOPGEN_4:6</a></span>;<br/>
<a NAME="E7:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#D6" onmouseover="rs('topmetr/D6')" onmouseout="rh()">TOPMETR:def 6</a>, <a class="ref" href="waybel23.html#D6" onmouseover="rs('waybel23/D6')" onmouseout="rh()">WAYBEL23:def 6</a></span>;<br/>
<a NAME="E8:48"/>
 <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 <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> #), <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> #) <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T77" onmouseover="rs('topreal6/T77')" onmouseout="rh()">TOPREAL6:77</a>, <a class="ref" href="topreala.html#T15" onmouseover="rs('topreala/T15')" onmouseout="rh()">TOPREALA:15</a></span>;<br/>
<a NAME="E9:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</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 <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#T4" target="_self" onmouseover="rs('metrizts/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E10:48"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #),<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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #)</span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#T6" target="_self" onmouseover="rs('metrizts/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E11:48"/><span class="kw">then </span>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E32">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:48"/>
 <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #) is  <a href="waybel23.html#V5" title="WAYBEL23:attr.5">second-countable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#D6" onmouseover="rs('waybel23/D6')" onmouseout="rh()">WAYBEL23:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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