Zero-dimensional space

in the sense of $\mathrm{ind}$

A topological space having a base of sets that are at the same time open and closed in it. Every discrete space is zero-dimensional, but a zero-dimensional space need not have isolated points (an example is the space $\mathbf{Q}$ of rational numbers). All zero-dimensional spaces are completely regular. Zero-dimensionality is inherited by subspaces and implies total disconnectedness of the space: The only connected sets in a zero-dimensional space are the singletons and the empty set. However, the latter property is not equivalent to being zero-dimensional. There are spaces that are not zero-dimensional and in which every point is the intersection of a family of open-and-closed sets, but no such space can be compact.

Sometimes the zero-dimensionality of a space is understood more narrowly. A space is called zero-dimensional in the sense of $\mathrm{dim}$ if every finite open covering of it can be refined to an open covering with disjoint elements. A space is called zero-dimensional in the sense of $\mathrm{Ind}$ if any neighbourhood of any closed subset of it contains an open-and-closed neighbourhood of this subset. In the class of $T_1$-spaces, zero-dimensionality in the sense of $\mathrm{ind}$ follows from both that in the sense of $\mathrm{dim}$ and that in the sense of $\mathrm{Ind}$. In the class of metrizable spaces with a countable base, and also in the class of Hausdorff compacta, the three definitions of being zero-dimensional are equivalent. For all metrizable spaces, zero-dimensionality in the sense of $\mathrm{dim}$ is equivalent to that in the sense of $\mathrm{Ind}$; however, an example is known of a metrizable space that is zero-dimensional in the sense of $\mathrm{ind}$, but not in the sense of $\mathrm{Ind}$. Neither zero-dimensionality in the sense of $\mathrm{Ind}$ nor that in the sense of $\mathrm{dim}$ is inherited by subspaces. Among $T_1$-spaces the zero-dimensional ones in the sense of $\mathrm{ind}$ can be characterized, up to a homeomorphism, as subspaces of generalized Cantor discontinua $D^\tau$ — products of colons. Any completely-regular space can be obtained as the image of a zero-dimensional space under a well-behaved mapping, for example, under a perfect mapping and under a continuous open mapping with compact inverse images of points. However, continuous mappings that are simultaneously open and closed preserve zero-dimensionality in the sense of $\mathrm{ind}$ and of $\mathrm{Ind}$. It is not known whether every completely-regular space contains an everywhere-dense zero-dimensional subspace.

References

[1]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)

Comments

A set that is simultaneously open and closed is sometimes called a clopen set.

The $T_0$ separation axiom is frequently assumed (without explicit mention) as part of the definition of zero-dimensionality: indeed, some of the assertions in the main article above are false if this assumption is not made.

Zero-dimensionality in the sense of $\mathrm{Ind}$ and $\mathrm{dim}$ are equivalent. This type of zero-dimensionality is generally called strong zero-dimensionality. It is clear that strongly zero-dimensional spaces are normal. A normal space $X$ is strongly zero-dimensional if and only if its Stone–Čech compactification $\beta X$ is zero-dimensional (in any sense). This fact motivates the extension of the notion of strong zero-dimensionality to the class of completely-regular spaces: A completely-regular space $X$ is defined to be strongly zero-dimensional if its compactification $\beta X$ is zero-dimensional.

Every topological space is an open quotient of a zero-dimensional paracompact space [a4]. Completely-regular spaces are perfect images even of extremally-disconnected spaces.

A fourth definition of dimension agreeing with the basic three in separable metric spaces is Krull dimension $\mathrm{gdim}$; see [a5] and (the editorial comments to) Lattice. For general metric spaces $X$, $\mathrm{ind}(X) \le \mathrm{gdim}(X) \le \mathrm{dim}(X)$ [a5].

For completely-regular spaces one may define zero-dimensional spaces that map onto it in many ways. One important such construction is that of the absolute, or Gleason cover. Initially, A.M. Gleason constructed $\alpha Y$ for a compact Hausdorff space $Y$, and showed that it is the projective covering of $Y$ in a sense dual to that of injective hull (see Injective module) [a3]. In fact, applied to zero-dimensional compact Hausdorff spaces (also called Boolean spaces or Stone spaces, cf. Stone space), the construction is dual by Stone duality (see Boolean algebra) to the MacNeille completion of Boolean algebras, which is an instance of injective hulls (cf. Completion, MacNeille (of a partially ordered set)); equipped with appropriate "compatible" orderings, they also provide a duality (Priestly duality) for arbitrary distributive lattices. The Gleason cover has been considerably generalized: to $T_0$-spaces [a2], to topological algebras of general type [a1], and to toposes [a6].

The question at the end of the article has been solved partially: Assuming the Continuum hypothesis one can construct a space such that no uncountable subspace of it is zero-dimensional and such that no countable subset is dense.

References

[a1]	B. Banaschewski, "Projective covers in categories of topological spaces and topological algebras" J. Novák (ed.) et al. (ed.) , General topology and its relations to modern analysis and algebra (Proc. Kanpur, 1968) , 3 , Academia (1971) pp. 63–91
[a2]	A. Błaszczyk, "Extremally disconnected resolutions of $T_0$ spaces" Colloq. Math. , 32 (1974) pp. 57–68
[a3]	A. Gleason, "Projective topological spaces" Ill. J. Math. , 2 (1958) pp. 482–489
[a4]	J. Isbell, "A note on complete closure algebras" Math. Systems Theory , 3 (1969) pp. 310–312
[a5]	J.R. Isbell, "Graduation and dimension in locales" I.H. James (ed.) E.H. Kronheimer (ed.) , Aspects of Topology: in Memory of Hugh Dowker , Lect. notes London Math. Soc. , 93 , Cambridge Univ. Press (1985) pp. 195–210
[a6]	P.T. Johnstone, "The Gleason cover of a topos I" J. Pure Appl. Alg. , 19 (1980) pp. 171–192
[a7]	P.T. Johnstone, "The Gleason cover of a topos II" J. Pure Appl. Alg. , 22 (1981) pp. 229–247
[a8]	K. Ciesielski, "$L$-space without any uncountable $0$-dimensional subspace" Fundam. Math. , 125 (1985) pp. 231–235
[a9]	R. Engelking, "General topology" , Heldermann (1989)
[a10]	R. Engelking, "Dimension theory" , North-Holland (1978) (Translated from Polish)
[a11]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948) ((Appendix by L.S. Pontryagin and L.G. Shnirel'man in Russian edition.))
[a12]	P. Nyikos, "A survey of zero-dimensional spaces" S.P. Franklin (ed.) et al. (ed.) , Topology (Proc. 9th Annual Spring Conf. Memphis, 1975) , M. Dekker (1976) pp. 87–114
[a13]	P.T. Johnstone, "Stone spaces" , Cambridge Univ. Press (1983)