draft for something random. I've yet to figure out why the TODO part is needed.

topology.md

Yet another alternative axiomization of a Hausdorff topological space

(or: why does the intuitive axiomation of topological space makes sense)

For Hausdorff space, there's the following alternative definition of a topology, rather than based on "open sets" (typical definition), or touch-axiomization as in this MathOverflow post. (nevertheless, this definition is really close to that.)

Let $A$ be a set. Define an operator $\lim: A^ω → A∪{ε}$ (assume $ε$ is an abstract element not in $A$), which takes an infinite sequence of elements in $A$, and returns an element of $A$ if the sequence converges in $A$, or return $ε$ otherwise.

(observe that if the space is not Hausdorff, it may be the case that a sequence "converges" to multiple points.)

We define the following:

• A set $B ⊆ A$ is closed in $A$ if, for all sequence of $B$-elements ${b_i}$, then $lim{b_i} ∈ B$ or $lim{b_i}=ε$.
• A set $B ⊆ A$ is open in $A$ if, for all element $b ∈ B$ and sequence of elements ${c_i}$ in $A ∖ B$, then $lim{b_i}≠ b$.

Note that by this definition, a set is open if and only if its complement is closed. Also, $∅$ and $A$ are obviously both open and closed.

• The closure of a set $B ⊆ A$ is the set of all point in $A$ that is a limit of a sequence of points in $B$.

We want the closure of $B$ to contain $B$, which means the limit operator must satisfy that the limit of a constant sequence is the same point, in other words, for all $a ∈ A$, $\lim {a} = a$.

We want taking the closure to be an idempotent operation, so we ... (TODO?)

Now, we prove that the intersection of arbitrary family of closed sets is closed and union of two closed sets are closed:

• let $I=⋂_{i ∈ I} F_i$ where $I$ is an indexing set and each $F_i$ is closed. Then, for any sequence of points ${i_n}$ in $I$, each point belong to each $F_i$, so the limit of the sequence, if it exists, also belong to each $F_i$, thus belong to $I$.
• let $I=F ∪ F'$, where $F$ and $F'$ are closed. For any sequence of points ${i_n}$ in $I$ that has a limit, there is either an infinite subsequence in $F$, or there's an infinite subsequence in $F'$, either way because $F$ and $F'$ are closed we conclude that the sequence has a limit in $I$.

The second point above requires another axiom for the $\lim$ operation: if a sequence has a limit $≠ ε$, any subsequence of that sequence must have the same limit.