§ Defining continuity covariantly

• 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