Definition of upper hemicontinuity of a correspondence.

Question

When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity.

A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping.

One way to define upper hemicontinuity of $f$ is as below.

Given such a $f$, $f$ is said to be upper hemicontinuous at $\bar{x}$ if for every open set $V$ such that $f(\bar{x})\subset V$, there exists an open set $U$ such that $\bar{x}\in U$ and $x\in U$ implies $f(x)\subset V$.

However, I once read a different definition of upper hemicontinuity on a textbook:

Given such a correspondence $f$, $f$ is said to be upper hemicontinuous at $\bar{x}$ if for every sequence $\{x_n\}\rightarrow\bar{x}$ and for every open set $V$ such that $f(\bar{x})\subset V$, there exists an $N\in\mathbb{Z}$ such that $f(x_n)\subset V,~\forall n\geq N$.

Can someone help me prove that these two definitions are equivalent?

(It's easy to prove from the former to the latter, but what about the other way?)

Answer 1

Given an open set $V\subset Y$, denote $\widetilde V =\{x: f(x)\subset V\}$.

The first definition says: if $\bar x\in \widetilde V$, then $\bar x$ is an interior point of $\widetilde V$.

The second definition says: if $\bar x\in \widetilde V$ and $x_n\to x$, then $x_n\in \widetilde V$ for all large $n$.

Now all multi-valuedness disappears from the view: we are staring at the sequential characterization of interior points. Indeed, the following three are equivalent in first-countable topological spaces:

1. $x$ is an interior point of $A$.
2. $x$ is not in the closure of $A^c$.
3. $x$ is not the limit of any sequence in $A^c$.