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Concept# Clopen set

Summary

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "door spaces" their name.
Examples
In any topological space X, the empty set and the whole space X are both clopen.
Now consider the space X

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In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological

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In mathematics, an open set is a generalization of an open interval in the real line.
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In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its

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Let Omega subset of R-n be an open set, A is an element of R-nxn and G : Omega -> R-nxn be given. We look for a solution u : Omega -> R-n of the equation A del u + (del u)(t) A = G We also study the associated Dirichlet problem. (C) 2020 Elsevier Ltd. All rights reserved.

Bernard Dacorogna, Olivier Kneuss

Let n > 2 be even; r >= 1 be an integer; 0 < alpha < 1; Omega be a bounded, connected, smooth, open set in R-n; and nu be its exterior unit normal. Let f, g is an element of C-r,C-alpha((Omega) over bar; Lambda(2)) be two symplectic forms (i.e., closed and of rank n) such that f-g is orthogonal to the harmonic fields with vanishing tangential part, nu boolean AND f,nu boolean AND g is an element of C-r+1,C-alpha(partial derivative Omega; Lambda(3)) and nu boolean AND f = v boolean AND g on partial derivative Omega. Moreover assume that tg + (1-t)f has rank n for every t is an element of [0, 1]. We will then prove the existence of a phi is an element of Diff(r+1,alpha)((Omega) over bar; (Omega) over bar )satisfying

2011