nLab punctured neighborhood

Redirected from "punctured neighborhoods".

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 Definition

In classical mathematics, let (X,𝒪(X))(X, \mathcal{O}(X)) be a topological space. A punctured neighborhood of an element xXx \in X is the subset of a neighborhood UU of xx consisting of all elements in UU which are not equal to xx, or equivalently, the relative complement U/{x}U / \{x\}.

In constructive mathematics, let (X,𝒪(X))(X, \mathcal{O}(X)) be a topological space with a tight apartness relation #\#. An element xXx \in X is strictly apart from a subset UXU \subseteq X if for all elements yXy \in X, yUy \in U implies that x#yx \# y. A punctured neighborhood of an element xXx \in X is the subset of a neighborhood UU of xx consisting of all elements in UU which are strictly apart from xx.

The tight apartness relation is needed in constructive mathematics because in the real numbers, a neighborhood around a real number cc is an open interval (cϵ,c+ϵ)(c - \epsilon, c + \epsilon) where ϵ\epsilon is a positive real number, and a punctured neighborhood is the set of real numbers xx in the open interval (cϵ,c+ϵ)(c - \epsilon, c + \epsilon) where 0<|xc|0 \lt {|x - c|}. The tight apartness relation x#cx \# c holds precisely if and only if 0<|xc|0 \lt {|x - c|}.

 See also

Created on November 5, 2023 at 15:20:46. See the history of this page for a list of all contributions to it.