Why are topological spaces defined in terms of open sets, and not in terms of connected open sets

general-topology

I'm starting to learn about point-set topology, and I find the definition of topological spaces and open sets to be very weird. Since we care about continuity in topology, it's odd that topological spaces are defined based on open sets, which can be disconnected. It makes more sense that they should be built on connected sets.

Below is an alternative formulation I made of topological spaces:

A topological space is a set $X$ of points, and the set $\tau$ of all
"bunches" in $X$. All bunches are subsets of $X$. A topological space
must have the following properties:

  • The empty set and $X$ are both bunches.
  • Given a non-empty collection of bunches, if their intersection is non-empty, then their union is also a bunch.
  • Given two bunches, their intersection is also a bunch.

My definition of bunch is intended to be equivalent to connected open
sets. Open sets can be then defined as the union of disjoint bunches.

Is my alternative formulation equivalent to the usual formulation? If yes, why is this formulation not used? If no, then what's the difference?

Note: I'm told that my formulation seems related to bases, although I'm not sure what the exact relationship is.

Note 2: My initial intuition was to define continuous functions as functions that sends bunches to bunches, but that doesn't quite work.

Best Answer

One is of course free to define everything, but I do not think that the concept of a "bunch topology" is useful. It seems that you imagine a bunch to be a a connected open set. Since you require $X$ to be a bunch, you would restrict to connected spaces, but why not?

Let us now have a look at $X = \mathbb R^2$: What would be a bunch topology on $X$?

I think to obtain something useful we should regard each open disk $B_r(x)$ with radius $r$ around $x \in X$ as a bunch. Then $U = B_{1+\epsilon}(0,0) \cup B_{1+\epsilon}(0,2)$ is a bunch for each $\epsilon > 0$, similarly $V = B_{1+\epsilon}(2,0) \cup B_{1+\epsilon}(2,2)$ is a bunch. Hence also $U \cap V$ is a bunch. But for sufficiently small $\epsilon$ this set is not connected. This shows that your axioms do not produce the result you want.

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