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On Cleavability of Topological Spaces over R, R n , and R ω
Author(s) -
ARHANGEL ALEXANDER V.
Publication year - 1992
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1992.tb32243.x
Subject(s) - subspace topology , space (punctuation) , topological space , combinatorics , regular space , mathematics , isolated point , physics , topology (electrical circuits) , discrete mathematics , topological vector space , mathematical analysis , computer science , operating system
A space X is said to be cleavable over a space Y if for every subset A of X there exists a continuous mapping f : X → Y such that A = f −1 f(A) . It is shown that a compact space X is cleavable over the space R of real numbers if and only if X is homeomorphic to a subspace of R . It is also proved that if a space X is cleavable over R n , where n is a natural number, and V is a subspace of X homeomorphic to R n , then V is open in X . Every infinite connected subset of any space cleavable over R is shown to have a nonempty interior.