# Self-based space

From Topospaces

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.*

## Contents

## 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.