2 Changed ‘closed’ to ‘open’ in #3.

Here's how I like to think about it.

We can all agree that a topological space should be a set $X$ together with some extra structure encoding how the points of $X$ fit together. It seems pretty reasonable ask that this structure is sophisticated enough to answer the following question whenever $x \in S \subset X$:

Given any choice of "direction" is there freedom to nudge $x$ some small "amount" in that direction without bumping into any points of $X \setminus S$?

We say that $x$ is an interior point of $S$ if the answer to the above question is "yes". I would say the following assertions about interior points are completely reasonable.

1. Any $x \in X$ is an interior point of $X$.
2. If $S \subset T \subset X$ and $x$ is an interior point of $S$, then $x$ is an interior point of $T$.
3. If $S,T \subset X$ and $x$ is an interior point of both $S$ and $T$, then $x$ is an interior point of $S \cap T$.

For instance, (1) holds because there are no points in $X \setminus X$ to concern ourselves about bumping into. (3) holds because, if I specify a direction, then I can move $x$ an amount $a_s$ (in this direction) without hitting points from $X \setminus S$ and an amount $a_t$ without hitting points from $X \setminus T$, so if I move $x$ the smaller of these two amounts, I won't hit anything in $X \setminus (S \cap T)$.

If we take the above as axioms for a machine that tells us which points are interior to which sets, and then define an open set to be a set each of whose points is an interior point, then it is simple to recover the standard axioms for open sets:

1. $\varnothing$ and $X$ are open.
2. The union of arbitrarily many open sets is open.
3. The intersection of two closed open sets is closedopen.

The only issue I can see with this approach is that one might be able to convince oneself that interior points should satisfy more axioms. For instance, if $X = {0,1}$ and $1$ can be moved a little bit in any direction without bumping into $0$, then shouldn't it be possible to move $0$ a little bit in any direction without bumping into $1$? This would seem to preclude the existence of the Sierpiński topology ${\varnothing, {1} ,X}$. Or perhaps this is merely an invitation to be more imaginitive about the geometry of the situation? For instance, maybe there is a little round bowl with $1$ at the bottom and $0$ is sitting on the rim. If I give a $0$ a little push in the direction of $1$, no matter how small, $0$ will roll into the bowl and bump into $1$.

Here's how I like to think about it.

We can all agree that a topological space should be a set $X$ together with some extra structure encoding how the points of $X$ fit together. It seems pretty reasonable ask that this structure is sophisticated enough to answer the following question whenever $x \in S \subset X$:

Given any choice of "direction" is there freedom to nudge $x$ some small "amount" in that direction without bumping into any points of $X \setminus S$?

We say that $x$ is an interior point of $S$ if the answer to the above question is "yes". I would say the following assertions about interior points are completely reasonable.

1. Any $x \in X$ is an interior point of $X$.
2. If $S \subset T \subset X$ and $x$ is an interior point of $S$, then $x$ is an interior point of $T$.
3. If $S,T \subset X$ and $x$ is an interior point of both $S$ and $T$, then $x$ is an interior point of $S \cap T$.

For instance, (1) holds because there are no points in $X \setminus X$ to concern ourselves about bumping into. (3) holds because, if I specify a direction, then I can move $x$ an amount $a_s$ (in this direction) without hitting points from $X \setminus S$ and an amount $a_t$ without hitting points from $X \setminus T$, so if I move $x$ the smaller of these two amounts, I won't hit anything in $X \setminus (S \cap T)$.

If we take the above as axioms for a machine that tells us which points are interior to which sets, and then define an open set to be a set each of whose points is an interior point, then it is simple to recover the standard axioms for open sets:

1. $\varnothing$ and $X$ are open.
2. The union of arbitrarily many open sets is open.
3. The intersection of two closed sets is closed.

The only issue I can see with this approach is that one might be able to convince oneself that interior points should satisfy more axioms. For instance, if $X = {0,1}$ and $1$ can be moved a little bit in any direction without bumping into $0$, then shouldn't it be possible to move $0$ a little bit in any direction without bumping into $1$? This would seem to preclude the existence of the Sierpiński topology ${\varnothing, {1} ,X}$. Or perhaps this is merely an invitation to be more imaginitive about the geometry of the situation? For instance, maybe there is a little round bowl with $1$ at the bottom and $0$ is sitting on the rim. If I give a $0$ a little push in the direction of $1$, no matter how small, $0$ will roll into the bowl and bump into $1$.