A Universe of Sorts

Real analysis: coavriant definition: $f(\lim x) = \lim (f x)$ . Contravariant definition in analysis/topology: $f^{-1}(open)$ is open.
Contravariant in topology via sierpinski: $U \subseteq X$ is open iff characteristic function $f(x) = \begin{cases} T & x \in U \\ \bot & \text{otherwise} \end{cases}$ is continuous.
A function $f: X \to Y$ is continuous iff every function $f \circ s$ is continuous for every continuous $s: Y \to S$ . That is, a function is continuous iff the pullback of every indicator is an indicator.
A topological space is said to be sequential iff every sequentially open set is open.
A set $K \subseteq X$ is sequentially open iff whenever a sequence $x_n$ has a limit point in $K$ , then there is some $M$ such that $x_{\geq M}$ lies in $K$ . [TODO: check ]
Now consider $\mathbb N_\infty$ , the one point compactification of the naturals. Here, we add a point called $\infty$ to $\mathbb N$ , and declare that sets which have a divergent sequences and $\infty$ in them are open.
More abstractly, we declare all sets that are complements of closed and bounded sets with infinity in them as open. So a set $U \subseteq \mathbb N_{\infty}$ is bounded iff there exists a closed bounded $C \subseteq \mathbb N$ such that $U = \mathbb N / C \cup \{ infty \}$ .
A function $x: \mathbb N_\infty to X$ is continuous [wrt above topology ] iff the sequence $x_n$ converges to the limit $x_\infty$ .
See that we use functions out of $\mathbb N_\infty$ [covariant ] instead of functions into $S$ [contravariant ].
Now say a function $f: X \to Y$ is sequentially continuous iff for every continuous $x: \mathbb N_\infty \to X$ , the composition $f \circ x: \mathbb N_\infty \to Y$ is continuous. Informally, the pushforward of every convergent sequence is continuous.
Can show that the category of sequential spaces is cartesian closed .
Now generalize $\mathbb N_\infty$
https://twitter.com/EscardoMartin/status/1444791065735729155

§ Defining Continuity Covariantly