Self-based space

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This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.

Definition

A topological space is said to be self-based if it has a basis of open sets for which every basis set is homeomorphic to the whole space.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

Box products

This property of topological spaces is a box product-closed property of topological spaces: it is closed under taking arbitrary box products
View other box product-closed properties of topological spaces

Hereditariness on open subsets

This property of topological spaces is hereditary on open subsets, or is open subspace-closed. In other words, any open subset of a topological space having this property, also has this property

An open subset of a self-based space is self-based under the induced subspace topology. The idea is that we can choose a smaller basis element about each point in the subspace, that is homeomorphic to the open subset.