Topological space lacking a point

Question

We bestow X and Y with topologies $\tau_x,\tau_y$ for a certain continuous function $f$ from $X/\{x\}$ (with subspace topology) to $Y$.

Under which circumstances can we extend $f$ on all of $X$, $x$ included so that:

1) the new extension $f^*:X \rightarrow Y$ (which is equal to $f$ on $X/\{x\}$ ) is continuous with respect to $\tau_x$ and $\tau_y$

AND (the tricky part)

2) Without resorting to the metric notion of limit (I am restating the obvious because I mentioned no norm before; if there is a limit then an extension exists but is that both a necessary and sufficient condition?)

The point is that I am trying to see how I can define an extension on a topological space, the only way I know of is in a metric space.

Answer 1

$f^*:X\to Y$ is continuous iff whenever $A\subset X$ and $p\in Cl_X(A)$ then $f(p)\in Cl_Y(f^*(A)).$ So there are two conditions on $f$ that are necessary & sufficient for a continuous extension $f^*$ to exist: There must exist $y=f^*(x)\in Y$ such that

(1) whenever $x'\in Cl_X(\{x\})$ then $f(x')\in Cl_Y(\{y\}),$ and

(2) whenever $x\not \in S\subset X$ and $x\in Cl_X(S)$ then $y\in Cl_Y(f(S)).$

Remark: The usual topological def'n of continuity of $f^*:X\to Y$ is that $(f^*)^{-1}(V)$ is open in $X$ whenever $V$ is open in $Y$. My first sentence above is one of many statements that are equivalent to this def'n.