Tube lemma

The tube lemma states that if a compact space’s product with a point lies inside an open subset of a product space, a neighborhood of that point times the whole compact space also lies inside the open set.

Compact space: A topological space in which every open cover has a finite subcover. A finite subcover lets local product neighborhoods combine into one neighborhood covering the compact factor.
Projection map: A map from a product space to one factor that sends each tuple to its selected coordinate. The tube lemma helps prove that projections from products with compact factors are closed maps.
Cartesian product: The set of ordered pairs formed by choosing one element from each of two sets. The lemma concerns open subsets of a product formed from a compact space and another space.
Alexander subbase theorem: A criterion stating that a space is compact if every cover by members of a subbase has a finite subcover. It offers an alternative compactness proof technique, while the tube lemma uses a finite cover of one slice.
Open cover: A collection of open sets whose union contains a specified space. The proof covers the compact slice using product neighborhoods drawn from the given open set.
Closed map: A function between topological spaces that sends closed sets to closed sets. A tube around each point outside a projected closed set establishes that the projection’s complement is open.
Open set: A member of a topology, characterized by its inclusion in that topology. The lemma begins with an open set containing the entire product slice at one point.
Heine–Borel theorem: The theorem that a subset of Euclidean space is compact exactly when it is closed and bounded. It gives a metric-space criterion for compactness, whereas the tube lemma is purely topological.
Finite subcover: A finite subcollection of a cover that still covers the space. Compactness supplies finitely many neighborhoods, whose intersection works at the point.
Tychonoff theorem: The theorem that an arbitrary product of compact spaces is compact in the product topology. Tube-lemma arguments appear in proofs connecting compact factors and compact product spaces.

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