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François G. Dorais
François G. Dorais
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Def: Suppose X is topological space and B is a base for it. We say, that B is normal base, if following properties hold:

a. For any x∈X and A∈B, with x∈A, there exist A′∈B, such that x∉A′ and A∪A′=X.

b. If U and V are open sets from B, such that U∪V=X, than there exist U′ and V′, disjoint sets from B, such that X∖U⊆V′ and X∖V⊆U′.

It is well known that the space is completely regular iff it has normal base. This characterizations first published in Orrin Frink's paper "compactifications and semi: Frink, Orrin, Compactifications and semi-normal spaces, Am. J. Math. 86, 602-normal spaces"607 (1964). ZBL0129.38101.

It's easy part to prove that every completely regular space has normal base, because the set of co-zero sets is such a base, which is well-known to be a base for a completely regular space.

Inverse part, that every space which has normal base is completely regular, maybe is also easy, because there is a hint that this is actually the rewriting of proof of Urysohn lemma. But, however I still have a problem with this proof and can't construct the scheme like from Urysohn's lemma. Please, help me if you can.

Def: Suppose X is topological space and B is a base for it. We say, that B is normal base, if following properties hold:

a. For any x∈X and A∈B, with x∈A, there exist A′∈B, such that x∉A′ and A∪A′=X.

b. If U and V are open sets from B, such that U∪V=X, than there exist U′ and V′, disjoint sets from B, such that X∖U⊆V′ and X∖V⊆U′.

It is well known that the space is completely regular iff it has normal base. This characterizations first published in Orrin Frink's paper "compactifications and semi-normal spaces".

It's easy part to prove that every completely regular space has normal base, because the set of co-zero sets is such a base, which is well-known to be a base for a completely regular space.

Inverse part, that every space which has normal base is completely regular, maybe is also easy, because there is a hint that this is actually the rewriting of proof of Urysohn lemma. But, however I still have a problem with this proof and can't construct the scheme like from Urysohn's lemma. Please, help me if you can.

Def: Suppose X is topological space and B is a base for it. We say, that B is normal base, if following properties hold:

a. For any x∈X and A∈B, with x∈A, there exist A′∈B, such that x∉A′ and A∪A′=X.

b. If U and V are open sets from B, such that U∪V=X, than there exist U′ and V′, disjoint sets from B, such that X∖U⊆V′ and X∖V⊆U′.

It is well known that the space is completely regular iff it has normal base. This characterizations first published in: Frink, Orrin, Compactifications and semi-normal spaces, Am. J. Math. 86, 602-607 (1964). ZBL0129.38101.

It's easy part to prove that every completely regular space has normal base, because the set of co-zero sets is such a base, which is well-known to be a base for a completely regular space.

Inverse part, that every space which has normal base is completely regular, maybe is also easy, because there is a hint that this is actually the rewriting of proof of Urysohn lemma. But, however I still have a problem with this proof and can't construct the scheme like from Urysohn's lemma. Please, help me if you can.

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Erekle Khurodze
Erekle Khurodze
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O. Frink's characterization of completely regular spaces

Def: Suppose X is topological space and B is a base for it. We say, that B is normal base, if following properties hold:

a. For any x∈X and A∈B, with x∈A, there exist A′∈B, such that x∉A′ and A∪A′=X.

b. If U and V are open sets from B, such that U∪V=X, than there exist U′ and V′, disjoint sets from B, such that X∖U⊆V′ and X∖V⊆U′.

It is well known that the space is completely regular iff it has normal base. This characterizations first published in Orrin Frink's paper "compactifications and semi-normal spaces".

It's easy part to prove that every completely regular space has normal base, because the set of co-zero sets is such a base, which is well-known to be a base for a completely regular space.

Inverse part, that every space which has normal base is completely regular, maybe is also easy, because there is a hint that this is actually the rewriting of proof of Urysohn lemma. But, however I still have a problem with this proof and can't construct the scheme like from Urysohn's lemma. Please, help me if you can.