Dante space

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In mathematics, a Dante space is a type of topological space.

[edit] Definitions

Let X be a topological space; let Y be a topological subspace of X and let τ and λ be two infinite cardinal numbers. Y is said to be τ-monolithic in X if, for each A ⊆ Y such that |A| ≤ τ, the closure of A in X is a compact set of weight at most τ. X is said to τ-suppress Y if, whenever λ ≥ τ, A ⊆ Y and |A| ≤ exp(λ), it follows that there exists an A′ ⊆ X such that A is contained within the closure of A′ and |A′| ≤ λ. X is said to be a Dante space if, for every infinite cardinal τ, there exists an everywhere-dense subspace Y of X that is both τ-monolithic in itself and τ-suppressed by X.

[edit] Examples

• Every dyadic compactum is a Dante space.