Results 1 to 3 of 3

Thread: local bases help

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    166

    local bases help

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

    Prove that
    (1) If $\displaystyle F$ is a local base for $\displaystyle X$, then $\displaystyle \{-U: U \in F\}$ is a local base for $\displaystyle X$.
    (2) If $\displaystyle F$ is a local base at $\displaystyle x$, then both $\displaystyle \{U-x: U \in F\}$ and $\displaystyle \{x-U: U \in F\}$ are local bases for $\displaystyle X$.
    (3) if $\displaystyle F$ is a local base for $\displaystyle X$, then $\displaystyle \{U+a: a \in X, U \in F \}$ is a base for the topology on X.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by dori1123 View Post
    Prove that
    (1) If $\displaystyle F$ is a local base for $\displaystyle X$, then $\displaystyle \{-U: U \in F\}$ is a local base for $\displaystyle X$.
    (2) If $\displaystyle F$ is a local base at $\displaystyle x$, then both $\displaystyle \{U-x: U \in F\}$ and $\displaystyle \{x-U: U \in F\}$ are local bases for $\displaystyle X$.
    (3) if $\displaystyle F$ is a local base for $\displaystyle X$, then $\displaystyle \{U+a: a \in X, U \in F \}$ 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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Sep 2008
    Posts
    166
    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 $\displaystyle \mathbb{R}$, let $\displaystyle x = 0$, and if $\displaystyle U=(-1, 2)$ is an open set of $\displaystyle x$, then $\displaystyle -U=(-2, 1)$ is also an open set of $\displaystyle x$.
    $\displaystyle U-x$ is open set $\displaystyle U$ deleting $\displaystyle x$, I guess.
    I'm not sure about $\displaystyle x-U$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. local man, local min, inflection point problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 27th 2011, 11:44 PM
  2. Replies: 4
    Last Post: Mar 21st 2011, 01:23 AM
  3. Replies: 6
    Last Post: Jan 5th 2011, 02:34 AM
  4. determine local max or local min
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Nov 21st 2009, 08:13 PM
  5. local max and local min values
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 1st 2009, 11:11 AM

Search Tags


/mathhelpforum @mathhelpforum