# 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

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)
How to Cite This Entry:
Quasi-normal space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-normal_space&oldid=48390
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article