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Math Help - local bases help

  1. #1
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    local bases help

    Let X be a topological space.
    Let F = \{U: U is a neighborhood of x \in X\}
    Then F is a local base at x if for every neighborhood V of x, E \subset V, where E \in F.
    Also, F is a base if for every open set W \subset X, W = \cup G, where G \in F.

    Prove that
    (1) If F is a local base for X, then  \{-U: U \in F\} is a local base for X.
    (2) If F is a local base at x, then both \{U-x: U \in F\} and \{x-U: U \in F\} are local bases for X.
    (3) if F is a local base for X, then \{U+a: a \in X, U \in F \} is a base for the topology on X.
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Prove that
    (1) If F is a local base for X, then  \{-U: U \in F\} is a local base for X.
    (2) If F is a local base at x, then both \{U-x: U \in F\} and \{x-U: U \in F\} are local bases for X.
    (3) if F is a local base for X, then \{U+a: a \in X, U \in F \} is a base for the topology on X.
    Hi, dori.
    Can you plz give some explanations on belows in your question?
    -U, x-U, U-x, U+a
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  3. #3
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    Quote Originally Posted by aliceinwonderland View Post
    Hi, dori.
    Can you plz give some explanations on belows in your question?
    -U, x-U, U-x, U+a
    I'm not sure.
    In \mathbb{R}, let x = 0, and if U=(-1, 2) is an open set of x, then -U=(-2, 1) is also an open set of x.
    U-x is open set U deleting x, I guess.
    I'm not sure about x-U
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