Limits of sequences as continuous functions from certain topology on $\omega+1$ Let T be a topological space, $p\in T$ and $u_n : \mathbb{N}\to T$.
(Should be a well known fact): The point $p$ is a limit of $u_n$ iff $u_n^{-1}V$ is open for every neighbourhood $V$ of $p$, with cofinite topology on $\mathbb{N}$
(My thought): Can we extend the domain space such that such pairs $(u_n, p)$ correspond to continuous functions?
Consider the set $\omega +1$ with the topology consisting of empty set and $C \cup \{\omega\}$ for every cofinite $C$ in $\omega$.
Then a continuous function corresponds to a convergent $u_n$ and a limit of it at $u_\omega$, as should be easy to see.
Is this correct? Does this topology have a standard name? Is this part of something more general? On an intuitive side, seems strange to force $\omega$ to be in every set with a "tail", since in other spaces like extended real line we don't do this, which makes me think there is a better way to do this.
 A: Yes, that is correct. The space $\omega+1$ has the order topology wrt the natural (ordinal) order where $\omega$ has its usual order $0 < 1< 2< \ldots$ and $n < \omega$ for all $n \in \omega$. Because $\omega$ is the maximal element of the ordered set its basic neighbourhoods are o  the form $(x,\omega]$ for $x < \omega$, so $x \in \omega$ so of the form $(n, \omega] = \{m \in \omega\mid m > n\} \cup \{\omega\}$, precisely the cofinite neighbourhoods you describe (tails, essentially).
The order topology for points of $\omega$ itself is discrete: $0$ has $[0,1), [0,2), \ldots$ as basic neighbourhoods (being the minimum element)< so that comes down to essentially $\{0\} = [0,1)$, while $n>0$ has a neighbourhood of interval form $(n-1,n+1)=\{n\}$ (true in $\omega$, not in $\Bbb R$ or $\Bbb Q$ of course), so all points of $\omega$ are isolated points, and any function/sequence defined on them is automatically continuous, and the special neighbourhoods of $\omega$ ensure that $n \to u_n, \omega\to p$ is only continuous as a function from $\omega+1$ to $X$ iff $\lim_n u_n = p$ (also true if the limit is not unique, as can happen).
$\omega+1$ is just a standard ordinal space in the order topology, but experienced topologists often call it "the convergent sequence", it's a sort of prototypical space for that for these reasons. In fact a space with "(non-trivial) convergent sequences" is one that contains a homeomorphic copy of $\omega+1$, e.g. Some famous examples (like $\omega^* := \beta \omega\setminus \omega$ are examples of large compact space without convergent sequences.
It's called countable Fort space on Wikipedia and in the book "Counterexamples in Topology" by Steen and Seebach. Or you can call it the one-point compactification of the countable discrete space if you prefer.
