Self-based space
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
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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
A finite product of self-based spaces is self-based. In fact, we can take as the new basis, the rectangles obtained from basis elements in each of the factors.
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.