Topological space
2009-10-02 7:19 am
Let X be a topological space. A subset U of X is said to be a neighbourhood of a point x of X if and only if U includes an open set to which x belongs. Prove that a subset A of X is open if and only if it is a neighbourhood of each of its points.
回答 (1)
2009-10-02 8:35 am
✔ 最佳答案
( => )A is open => A is a neighbourhood of each x in A(def. of neighbourhood)
( <= )
A is a neighbourhood of each x in A
=> A contain an open subset A(x) for all x.
Let B the union of A(x) for all x in A. Then
A contain B and B contain A, hence, A=B
ie. A is the union of open set, so, A is open. (Axiom of topological space)
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