# 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.