Discrete spaceFrom Topospaces Jump to: navigation, search 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 is an opposite of compactness Show DefinitionA discrete space is a topological space satisfying the following equivalent conditions:
Given any set, there is a unique topology on it making it into discrete space. This is termed the discrete topology. The discrete topology on a set is the finest possible topology on the set. Relation with other propertiesWeaker properties
Related propertiesCompactness is the opposite of discreteness in some sense. The only topological spaces that are both discrete and compact are the finite spaces. MetapropertiesProductsThis property of topological spaces is closed under taking arbitrary products A (finite?) direct product of discrete spaces is discrete. HereditarinessThis property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property. Any subspace of a discrete space is discrete under the induced topology. Retrieved from "https://topospaces.subwiki.org/w/index.php?title=Discrete_space&oldid=4429" Categories:
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