Open cover

An open cover of a set is a family of open sets whose union contains that set. Covers are used to formulate compactness and related results in topology.

Topological space: A set equipped with a collection of subsets designated as open, satisfying axioms for arbitrary unions and finite intersections. Open covers are defined using the open sets of a topological space.
Lebesgue number lemma: A theorem stating that an open cover of a compact metric space admits a positive scale such that every sufficiently small subset lies in one cover member. It converts an open cover into a uniform distance scale across a compact space.
Tube lemma: A theorem stating that if a product of a space and a compact space lies in an open set, a product neighborhood can be chosen around the first factor. Its proof uses finite subcovers to control an entire compact fiber.
Closed cover: A family of closed sets whose union contains a given space or subset. It replaces openness with closedness, and its covering consequences differ substantially.
Open set: A subset designated as open by a topology, or satisfying the corresponding openness condition in a familiar space. Every member of an open cover must be open in the relevant topology.
Paracompact space: A topological space in which every open cover has a locally finite open refinement. Paracompactness guarantees a controlled refinement even when no finite subcover exists.
Lebesgue covering dimension: A topological dimension defined through the smallest multiplicity achievable by refinements of open covers. It measures dimension by how complicated suitably refined covers must remain.
Locally finite family: A family of sets for which every point has a neighborhood meeting only finitely many members. Locally finite refinements are a stronger structural demand than simply having a cover.
Union: The set of all elements belonging to at least one set in a given family of sets. A cover qualifies when the union of its members contains the target set.
Partition of unity: A family of continuous functions whose locally finite sum is one, often subordinate to an open cover. Partitions of unity translate a cover into functions supported in its open sets.

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