Suppose that $f : X \rightarrow Y$ is mapping between topological spaces that is not continuous at $x\_0$. Then there is an open set $V$ in $Y$ containing $f(x\_0)$ such that for any open set $U$ containing $x\_0$, there is some $x\_U \in U$ with $f(x\_U) \notin V$. By picking one from each $U$ we can build a net $x\_U$ converging to $x\_0$ such that $f(x\_U)$ does not converge to $f(x\_0)$. This however requires the axiom of choice because we have to pick $x\_U$ from each $U$. 


My question is: if we are given that a function between topological spaces is continuous if and only if it is "sequentially" continuous (in the sense of nets, not necessarily infinite sequences indexed by the integers) - does the axiom of choice follow?



Nets and the Axiom of Choice
The axiom of choice is not needed if $X$ is $T\_1$; i.e., singleton sets are closed.


Suppose $f:X\to Y$ is not continuous at $x\_0$. Let $D$ be the collection of all $(U,x)$ s.t. $U$ is an open set containing both $x\_0$ and $x$ and $x\_0\not=x$. Say that $(U,x) \le (V,y)$ provided $V\subset U$ and $x$ is not in $V$. If $X$ is $T\_1$, this is a directed set. Define a net $g$ with domaine $D$ by $g(U,x)=x$. Then $g$ converges to $x\_0$ but $f(g)$ does not converge to $f(x\_0)$.


