Defining Topologies: The Role of Partial Order in Point-Set Topology

[Shaggy16]

Homework Statement

I started studying point-set topology a while ago, and I started to wonder, "Does a set have to be partially ordered in order to define a topology on it?"

Homework Equations

The Attempt at a Solution

I know that every set in a topology has to be open, which means that at any point you can construct an open ball such that every point in the open ball is also in that set. I don't see how this would work unless there was some sort of relation like <, >, or something similar on the set. Perhaps I'm not seeing the generalization to sets other than R^n, but I'm at a complete loss.

 

[HallsofIvy]

A "topology" for a set, X, is any collection of subsets of X satisfying

1) Both X itself and the empty set is in the topology.

2) The union of any sub-collection of those subsets in the topology is in the topology.

3) The intersection of any finite subcollection of subsets in the topology is in the topology.

There is no need for a partial order or even "open balls" to define that.

For example, given any set, X, one possible topology is collection of all subsets of X, the "discrete" topology. Another is topology containing only the two sets {{}, X}, the "indiscrete" topology.

 

[Shaggy16]

Then where do open sets come into play?

 

[Shaggy16]

Okay, that was a really stupid question. An open set is just a set contained in the topology. If you are considering R^n, then it is implied that an open set is one such that at every point one could construct an open ball such that all of the points in the open ball are also in the set, because only open balls can be in the topology. Therefore, one has to have a sense of which elements are less than and which are greater than other elements in order to construct said ball, hence the partial order. Sorry, everybody...