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A proof for normal spectral spaces property
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Let be $(X,tau)$ a spectral space and $X_M$ the subspace of closed points of $X$.
A point $yin X$ is a specialisation of $xin X$ if $y$ is in the closure of $x$. I find this theorem:
if every $xin X$ has a unique specialisation $pi(x) in X_M$, then, $(X,tau)$ is normal.
I find a proof, but I don't understand a passage of it.
Let $F, G$ two non empty closed sets such that $Fcap G=emptyset$.
We consider the family $oversetcircmathcalH(X,tau)$ of open compact sets (a base for $tau$). We suppose, by absurdum, that $(X,tau)$ is not normal.
$forall V,Win oversetcircmathcalH(X,tau)qquad Fsubseteq Vquad wedgequad Gsubseteq WRightarrow Vcap Wne emptyset$.
We consider the family $mathscrF=leftVcap W:left$.
$mathscrF$ is a family of closed sets of $X$ with constructible topology, that is compact and Hausdorff; $mathscrF$ has finite intersection property, so $bigcapmathscrFne emptyset$. Let be $tinbigcapmathscrF$. Perhaps it is trivial, but how can I prove that $F$ and $G$ intersect the closure of $t$?
Thank you.
general-topology
asked Jul 23 at 8:13
Fabrixady
12
add a comment |Â
up vote
0
down vote
favorite
Let be $(X,tau)$ a spectral space and $X_M$ the subspace of closed points of $X$.
A point $yin X$ is a specialisation of $xin X$ if $y$ is in the closure of $x$. I find this theorem:
if every $xin X$ has a unique specialisation $pi(x) in X_M$, then, $(X,tau)$ is normal.
I find a proof, but I don't understand a passage of it.
Let $F, G$ two non empty closed sets such that $Fcap G=emptyset$.
We consider the family $oversetcircmathcalH(X,tau)$ of open compact sets (a base for $tau$). We suppose, by absurdum, that $(X,tau)$ is not normal.
$forall V,Win oversetcircmathcalH(X,tau)qquad Fsubseteq Vquad wedgequad Gsubseteq WRightarrow Vcap Wne emptyset$.
We consider the family $mathscrF=leftVcap W:left$.
$mathscrF$ is a family of closed sets of $X$ with constructible topology, that is compact and Hausdorff; $mathscrF$ has finite intersection property, so $bigcapmathscrFne emptyset$. Let be $tinbigcapmathscrF$. Perhaps it is trivial, but how can I prove that $F$ and $G$ intersect the closure of $t$?
Thank you.
general-topology
asked Jul 23 at 8:13
Fabrixady
12
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let be $(X,tau)$ a spectral space and $X_M$ the subspace of closed points of $X$.
A point $yin X$ is a specialisation of $xin X$ if $y$ is in the closure of $x$. I find this theorem:
if every $xin X$ has a unique specialisation $pi(x) in X_M$, then, $(X,tau)$ is normal.
I find a proof, but I don't understand a passage of it.
Let $F, G$ two non empty closed sets such that $Fcap G=emptyset$.
We consider the family $oversetcircmathcalH(X,tau)$ of open compact sets (a base for $tau$). We suppose, by absurdum, that $(X,tau)$ is not normal.
$forall V,Win oversetcircmathcalH(X,tau)qquad Fsubseteq Vquad wedgequad Gsubseteq WRightarrow Vcap Wne emptyset$.
We consider the family $mathscrF=leftVcap W:left$.
$mathscrF$ is a family of closed sets of $X$ with constructible topology, that is compact and Hausdorff; $mathscrF$ has finite intersection property, so $bigcapmathscrFne emptyset$. Let be $tinbigcapmathscrF$. Perhaps it is trivial, but how can I prove that $F$ and $G$ intersect the closure of $t$?
Thank you.
general-topology
asked Jul 23 at 8:13
Fabrixady
12
Let be $(X,tau)$ a spectral space and $X_M$ the subspace of closed points of $X$.
A point $yin X$ is a specialisation of $xin X$ if $y$ is in the closure of $x$. I find this theorem:
if every $xin X$ has a unique specialisation $pi(x) in X_M$, then, $(X,tau)$ is normal.
I find a proof, but I don't understand a passage of it.
Let $F, G$ two non empty closed sets such that $Fcap G=emptyset$.
We consider the family $oversetcircmathcalH(X,tau)$ of open compact sets (a base for $tau$). We suppose, by absurdum, that $(X,tau)$ is not normal.
$forall V,Win oversetcircmathcalH(X,tau)qquad Fsubseteq Vquad wedgequad Gsubseteq WRightarrow Vcap Wne emptyset$.
We consider the family $mathscrF=leftVcap W:left$.
$mathscrF$ is a family of closed sets of $X$ with constructible topology, that is compact and Hausdorff; $mathscrF$ has finite intersection property, so $bigcapmathscrFne emptyset$. Let be $tinbigcapmathscrF$. Perhaps it is trivial, but how can I prove that $F$ and $G$ intersect the closure of $t$?
Thank you.
general-topology
asked Jul 23 at 8:13
Fabrixady
12
asked Jul 23 at 8:13
Fabrixady
12
asked Jul 23 at 8:13
Fabrixady
12
asked Jul 23 at 8:13
Fabrixady
12
12
add a comment |Â
add a comment |Â
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