1. ## 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.

2. Originally Posted by dori1123
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

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 $\mathbb{R}$, 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 $x-U$