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.

  • $\begingroup$ What extension? What metric? $\endgroup$ Dec 7 '17 at 22:34
  • $\begingroup$ @B.Pasternak I have edited the question for your convenience. $\endgroup$
    – Averroes
    Dec 7 '17 at 22:39
  • $\begingroup$ You cannot resort to the metric notion of limit if you have no metric. I also see no norms, so there are no norms to be omitted. I still don't get your question: you seem to be taking a continuous extension $f*$ of $f$, and then asking whether you can add $x$ to $X\setminus\{x\}$ without ruining the continuity of $f*$? What's the actual question, what are your precise hypotheses? $\endgroup$ Dec 7 '17 at 23:16
  • $\begingroup$ @B.Pasternak my question is general. I insisted on the lack of norm by repeating it literally. $f^*$ is rather the extension that I seek. In other terms, How can we extend $f$ to $x$ (thus defining $x$) the while we keep the continuity of the resulting extension. $\endgroup$
    – Averroes
    Dec 7 '17 at 23:20
  • $\begingroup$ in topological terms this time $\endgroup$
    – Averroes
    Dec 7 '17 at 23:28

$f^*:X\to Y$ 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.


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