Difference between revisions of "Interior"
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==Definition== | ==Definition== | ||
Given a set {{M|U\subseteq X}} and an arbitrary [[metric space]], {{M|(X,d)}} or [[topological space]], {{M|(X,\mathcal{J})}} the ''interior'' of {{M|U}}, denoted {{M|\text{Int}(U)}} is defined as{{rITTGG}}{{rITTBM}}: | Given a set {{M|U\subseteq X}} and an arbitrary [[metric space]], {{M|(X,d)}} or [[topological space]], {{M|(X,\mathcal{J})}} the ''interior'' of {{M|U}}, denoted {{M|\text{Int}(U)}} is defined as{{rITTGG}}{{rITTBM}}: | ||
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Definition
Given a set U⊆X and an arbitrary metric space, (X,d) or topological space, (X,J) the interior of U, denoted Int(U) is defined as[1][2]:
- Int(U):={x∈X| x is interior to U} - (see interior point for the definition of what it means to be interior to)
Note that, unlike interior point which is basically a synonym for neighbourhood (taking the definition of neighbourhood as discussed on its page) the interior is a meaningful and distinct definition. In accordance with the topological definition of interior point (requiring that U be a neighbourhood to some x∈X) we see that:
- Int(U) is the set of all points U is a neighbourhood to.
Immediate properties
Let U⊆X be an arbitrary subset of a topological space (X,J) (as all metric spaces are topological, they are included), then:
