1. ## 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.

2. Originally Posted by dori1123
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

3. Originally Posted by aliceinwonderland
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 $\displaystyle x-U$