replace convergence with continuity?(metric spaces)

Question

This question is convcerning metric-spaces. In theory we can replace continuity with convergence. That is, since continuity in a point a is equal to the statement that if $\{x_n\}$ is any sequence converging to a, then $\{f(x_n)\}$ converges to f(a), we could just use this definition, and never work with the continuity -definition.

But can we do the opposite in some way? Can we start with the continuity definition and do something so that we never have to use the convergence definition, only the continuity?

Answer 1

In principle, yes. The mapping $$ x \colon \mathbb{N}\cup\{\infty\} \to X, \qquad x(n)=\begin{cases} x_n, & n < \infty, \\ x_\infty, & n = \infty \end{cases}$$ is continuous if and only if $x_\infty = \lim_{n\to \infty} x_n$. But this (rather trivial) observation is not very useful in practice.

P.S.: A metric on $\mathbb{N}\cup \{\infty\}$ is the following: $$ d_\infty ( n_1, n_2)=\lvert \arctan(n_1)-\arctan(n_2)\rvert, $$ where $\arctan(\infty)=\pi/2$.

Answer 2

For the concept of continuity of a function, in general there are two ways to define it:

1. Using Neighbourhoods. A function is continuous at $a$ if for every neighbourhood of $f(a)$, say $V_{f(a)}$, there exists a neighbourhood of $a$, say $U_a$ such that whenever $x\in U_a$ we have $f(x) \in V_{f(a)}$. For example, the $\epsilon, \delta$ definition of continuity uses this .

2. Using Sequence. A function is continuous at $a$ if $\forall x_n \rightarrow a$ in the topology of the domain space, we have $f(x_n) \rightarrow f(a)$ in the topology of the range space.

We call 1 "continous" and 2 "sequential continous". In a first countable topological spaces, they are equivalent. And every metric space is a first countable topological space.