Zero-dimensional space

A topological space with a basis of sets that are both open and closed, called clopen sets. In the metrizable case, this is equivalent to having small inductive dimension zero.

Clopen set: A subset that is both open and closed in a topological space. Clopen sets form the basis that defines zero-dimensionality.
Cantor space: The product space of countably many copies of the discrete two-point space. Its finite-coordinate cylinders are clopen and form a basis.
Connected space: A topological space that cannot be partitioned into two nonempty disjoint open sets. A connected zero-dimensional space with a clopen basis must be a singleton.
Stone representation theorem: The theorem representing every Boolean algebra as the algebra of clopen sets of a compact Hausdorff space. It uses zero-dimensional compact Hausdorff spaces to realize Boolean algebras topologically.
Basis (topology): A collection of open sets whose unions generate all open sets of a topology. Zero-dimensionality requires such a basis to consist entirely of clopen sets.
Profinite space: An inverse limit of finite discrete spaces, typically compact and Hausdorff. Finite-stage conditions give a clopen basis.
Totally disconnected compact Hausdorff space: A compact Hausdorff space whose connected subsets are all singletons. In compact Hausdorff spaces, total disconnectedness is equivalent to zero-dimensionality.
Symbolic dynamics: The study of dynamical systems defined by shifts on sequences of symbols. Shift spaces often have clopen cylinder sets as a basis.
Small inductive dimension: A topological dimension defined recursively through boundaries of neighborhoods. For metrizable spaces, dimension zero is equivalent to having a clopen basis.
p-adic integers: The inverse limit of the rings of integers modulo powers of a prime p. Residue classes modulo powers of p are clopen neighborhoods.

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