In mathematics, a topological space is zero-dimensional or 0-dimensional, if its topological dimension or Lebesque covering dimension is zero, or, equivalently, every open covering of the space has a refinement which is a covering of the space by open sets such that any two distinct open sets of this refinement are disjoint.
Note that this is not equivalent to another notion of a zero-dimensional space, a space with small inductive dimension 0, in which a zero-dimensional space has a base consisting of clopen sets.
To see that this other notion of a zero-dimensional space is not equivalent to the notion of a topological space with Lebesque covering dimension 0, consider the following example.
Let (S,T) be the topological space with S = {a,b} and whose open sets (i.e. elements of T) are the empty set, {a}, and {a,b}.
On the one hand, since {a,b} is the only element of T containing b, every open covering C of the topological space (S,T) contains {a,b}. Hence, the singleton set D with a single element {a,b} is a refinement of C which is a covering of (S,T) by open sets such that any two distinct open sets of this refinement are disjoint (since there are not any two distinct open sets of this refinement). Thus, (S,T) has topological dimension 0 (i.e Lebesque covering dimension 0).
On the other hand, suppose that B is a base for (S,T) consisting of clopen sets in (S,T). Note that B is contained in T. Note that {a} is an open neighborhood of a in (S,T). Since, by assumption, B is a base for (S,T), there exists an element U of B such that a is an element of U and U is contained in {a}. Since B is contained in T and U is an element of B, it follows that U is an element of T. Thus, U is an element of T such that a is an element of U and U is contained in {a}. Note that {a} is the unique such element of T. It follows that U = {a}. Since U is an element of B (i.e. {a} is an element of B) and B consists of clopen sets in (S,T), it follows that {a} is a clopen set in (S,T). In particular, it follows that {a} is closed in (S,T). This implies that S - {a} = {a,b} - {a} = {b} is open in (S,T) (i.e. is an element of T). But {b} is not an element of T. This proves that (S,T) does not have a basis of clopen sets, even though it has topological dimension 0 (i.e. Lebesque covering dimension 0).
A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected.
Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space.
Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers 2I where 2={0,1} is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, 2I is the Cantor space.
References
- Ryszard Engelking (1977). General Topology. PWN, Warsaw.
- Stephen Willard (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
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