Quasi-normal space

A regular space in which two disjoint $ \pi $- sets have disjoint neighbourhoods. Every $ T _ \lambda $- space in which any two disjoint $ \pi $- sets have disjoint neighbourhoods is a quasi-normal space. Only for the quasi-normal spaces does the Stone–Čech compactification $ \beta X $ coincide with the space $ \omega _ \kappa X $. The following theorem provides a large supply of non-normal quasi-normal spaces: The product of any number of separated metric spaces is quasi-normal.

References

[1]	V.I. Zaitsev, "Projection spectra" Trans. Moscow Math. Soc. , 27 (1972) pp. 135–199 Trudy Moskov. Mat. Obshch. , 27 (1972) pp. 129–193
[2]	E.V. Shchepin, "On the bicompact Ponomarev–Zaicev extension and the so-called spectral parasite" Math. USSR Sb. , 17 (1972) pp. 317–326 Mat. Sb. , 88 : 2 (1972) pp. 316–325

Comments

Quasi-normal spaces arose in the study of the spectrum of a topological space (cf. also Spectrum of spaces). This spectrum is obtained as follows. A partition of a space $ X $ is a finite collection $ {\mathcal A} $ of canonical closed sets (cf. Canonical set) that covers $ X $ and the elements of which have disjoint interiors. The set of all these partitions is partially ordered by: $ {\mathcal A} \succ {\mathcal A} ^ \prime $ if and only if $ {\mathcal A} $ refines $ {\mathcal A} ^ \prime $. The nerve $ N _ {\mathcal A} $ of $ {\mathcal A} $( cf. Nerve of a family of sets) is the complex of subfamilies of $ {\mathcal A} $ that have a non-empty intersection. There is an obvious simplicial mapping $ \omega _ { {\mathcal A} ^ \prime } ^ {\mathcal A} : N _ {\mathcal A} \rightarrow N _ { {\mathcal A} ^ \prime } $ if $ {\mathcal A} \succ {\mathcal A} ^ \prime $. If the set of partitions of $ X $ is (upward) directed by $ \succ $, then the inverse spectrum $ \{ {N _ {\mathcal A} , \omega _ { {\mathcal A} ^ \prime } ^ {\mathcal A} } : { {\mathcal A} \textrm{ a partition of } X } \} $ is the spectrum of $ X $ and is denoted by $ S _ {X} $. To obtain a suitable limit of this spectrum one takes the set $ \widetilde{S} {} _ {X} $ of maximal threads of $ S _ {X} $. A thread is a choice $ \{ t _ {\mathcal A} \} _ {\mathcal A} $ of simplexes with $ t _ {\mathcal A} \in {\mathcal A} $ for all $ {\mathcal A} $ and such that $ \omega _ { {\mathcal A} ^ \prime } ^ {\mathcal A} ( t _ {\mathcal A} ) = {\mathcal A} ^ \prime $ whenever $ {\mathcal A} \succ {\mathcal A} ^ \prime $. A thread $ \mathbf t = \{ t _ {\mathcal A} \} _ {\mathcal A} $ is maximal if whenever $ \mathbf t ^ \prime = \{ t _ {\mathcal A} ^ \prime \} _ {\mathcal A} $ is another thread such that $ t _ {\mathcal A} $ is a face of $ t _ {\mathcal A} ^ \prime $ for every $ {\mathcal A} $, one has $ \mathbf t = \mathbf t ^ \prime $. The basic open sets are the sets of the form $ O( t _ {\mathcal A} ) = \{ {\mathbf t ^ \prime \in \widetilde{S} {} _ {X} } : {t _ {\mathcal A} ^ \prime \textrm{ is a face of } t _ {\mathcal A} } \} $.

The space $ \omega _ \kappa X $, first introduced in [a3], is the space of all maximal centred systems of canonical closed sets topologized in the usual way, i.e. by taking the collection $ \{ {A ^ {+} } : {A \textrm{ a canonical closed set } } \} $ as a base for the closed sets of $ \omega _ \kappa X $, where $ A ^ {+} $ is the set of maximal centred systems to which $ A $ belongs.

It turns out that there is a natural homeomorphism from $ \omega _ \kappa X $ onto $ \widetilde{S} {} _ {X} $. Thus, for quasi-normal spaces $ X $ one has $ \beta X = \widetilde{S} {} _ {X} = \omega _ \kappa X $.

A $ \pi $- set is a finite intersection of closures of open sets. A $ T _ \lambda $- space, first introduced in [1], is a semi-regular ( $ T _ {1} $-) space all open sets of which are unions of $ \pi $- sets. I.e., a $ T _ \lambda $- space is a semi-regular ( $ T _ {1} $-) space (the canonical open sets form a base for the topology) in which the canonical closed sets form a net (of sets in a topological space), i.e. if $ O $ is open and $ x \in O $, then there is a canonical closed set $ A $ such that $ x \in A \subseteq O $.

References

[a1]	A. Kurosh, "Kombinatorischer Aufbau der bikompakten topologischen Räume" Compositio Math. , 2 (1935) pp. 471–476
[a2]	V.I. Zaitsev, "Finite spectra of topological spaces and their limit spaces" Math. Ann. , 179 (1968–1969) pp. 153–174
[a3]	V.I. Ponomarev, "Paracompacta: their projection spectra and continuous mappings" Mat. Sb. , 60 (102) (1963) pp. 89–119 (In Russian)