TITLE: Does this property characterize a space as Hausdorff?
QUESTION [105 upvotes]: As a result of this question, I've been thinking about the following condition on a topological space $Y$:

For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense in $X$, and $f$ and $g$ agree on $E$ (that is, $f(e)=g(e)$ for all $e\in E$), then $f=g$.

If $Y$ is Hausdorff, then $Y$ satisfies this condition. The question is whether the converse holds: if $Y$ satisfies the above condition, will it necessarily be Hausdorff? 
If $Y$ is not at least $T_1$, then $Y$ does not have the property: if $u,v\in Y$ are such that $u\neq v$ and every open neighborhood of $u$ contains $v$, then let $X$ be the Sierpinski space, $X=\{a,b\}$, $a\neq b$, with topology $\tau=\{\emptyset,\{b\},X\}$, $E=\{b\}$, let $f,g\colon X\to Y$ be given by $f(a)=f(b)=v$, and $g(a)=u$, $g(b)=v$. Then both $f$ and $g$ are continuous, agree on the dense subset $E$, but are distinct.
My attempt at a proof of the converse assumes the Axiom of Choice and proceeded as follows: assume $Y$ is $T_1$ but not $T_2$; let $u$ and $v$ be witnesses to the fact that $Y$ is not $T_2$, let $\mathcal{U}\_s$ and $\mathcal{V}\_t$ be the collection of all open nbds of $s$ that do not contain $t$, and all open nbds of $t$ that do not contain $s$, respectively. Construct a net with index set $\mathcal{U}\_s\times\mathcal{V}\_t$ (ordered by $(U,V)\leq (U',V')$ if and only if $U'\subseteq U$ and $V'\subseteq V$) by letting $y_{(U,V)}$ be a point in $U\cap V$ (this is where AC comes in). Let $E=\{y_{(U,V)}\mid (U,V)\in\mathcal{U}\_s\times\mathcal{V}\_t\}$, and let $X=E\cup\{s\}$. Give $X$ the induced topology; let $f\colon X\to Y$ be the inclusion map, and let $g\colon X\to Y$ be the map that maps $E$ to itself identically, but maps $s$ to $t$. 
The only problem is I cannot quite prove that $g$ is continuous; the difficulty arises if I take an open set $\mathcal{O}\in \mathcal{V}_t$; the inverse image under $g$ is equal to $((\mathcal{O}\cap X)-\{t\})\cup\{s\}$, and I have not been able to show that this is open in $X$. 
So:

Does the condition above characterize Hausdorff spaces? 

If not, I would appreciate a counterexample. If it does characterize Hausdorff, then ideally I would like a way to finish off my proof, but if the proof is unsalvageable (or nobody else can figure out how to finish it off either) then any proof will do. 

Added: A little digging turned up  this question raised in the Problem Section of the American Mathematical Monthly back in 1964 by Alan Weinstein. The solution by Sim Lasher gives a one paragraph proof that does not require one to consider $T_1$ and non-$T_1$ spaces separately.

REPLY [13 votes]: This problem has a very straightforward solution when you conceptualize convergence of nets in terms of continuity of maps.  Given a directed set $I$, the statement that a net $(y_i)_{i\in I}$ in a space $Y$ converges to a point $y$ can be expressed in terms of continuity of a map.  Namely, let $X=I\cup\{\infty\}$, topologized such that a set is open iff either it does not contain $\infty$ or it contains the set $\{j\in I:j\geq i\}$ for some $i\in I$.  Then $(y_i)$ converges to $y$ iff the map $i\mapsto y_i$, $\infty\mapsto y$ is a continuous map from $X$ to $Y$.
Now we use the characterization of Hausdorff spaces in terms of nets: a space is Hausdorff iff every net has at most one limit.  So if $Y$ is not Hausdorff, there is some net $(y_i)_{i\in I}$ in $Y$ which converges to two distinct points $y$ and $y'$, which means that the map $I\to Y$ sending $i$ to $y_i$ can be extended continuously to $X$ by sending $\infty$ to either $y$ or $y'$.  Since $I$ is dense in $X$, this is exactly what we're looking for.