Why Are Open Sets Preferred in Defining Domains in Topology?

[pivoxa15]

I am doing a bit of complex analysis and there is a definition:

An open connected subset of C is called a domain.

Why do they choose open instead of closed? Why not include a closed connected subset of C be a domain also?

 

[ircdan]

Well it's just defined that way, but why you ask? Well here is my take on it. Perhaps others can provide more insight.

Recall that for the derivative of a function f(z) to exist at a point z_0, f(z) must be defined in a neighborhood of z_0. Now let D be an open set, then D has the nice property that for any point z_0 in D there is a neighborhood about z_0. Furthermore, a function f(z) is said to be analytic on an open set D if it is has a derivative at every point in D. If D was closed, I don't see how it could have a derivative at every point in D(there are right-hand and left-hand derivatives, but I don't know about these or if this will work). Complex analysis seems to be concerned with the theory of analytic functions, and so I think this is why it makes sense to define a domain as an open connected set.

The fact that it's connected is useful also. You will see this comes into play when you learn about Cauchy's Integral Theorem and lots of other stuff.

 

[matt grime]

Why should we look at closed things? Evidently someone observed that they kept coming up with situations theat required open connected sets so they chose a name for them. No similar need was found for closedness.

In any case, analyticity is an 'open' thing. Something is analytic at z if it is equal to its taylor series in some neighbourhood about that point. The radius of convegence defines an open connected domain about z on which it is analytic, and this is a maximal domain in some sense: there *is* a point on the boundary of the disc where the talyor series diverges: that is what the radius of convergence actually means.
So, analytic functions are gotten by patching information on open connected domains together.

 

[pivoxa15]

Why should we look at closed things? Evidently someone observed that they kept coming up with situations theat required open connected sets so they chose a name for them. No similar need was found for closedness.

In any case, analyticity is an 'open' thing. Something is analytic at z if it is equal to its taylor series in some neighbourhood about that point. The radius of convegence defines an open connected domain about z on which it is analytic, and this is a maximal domain in some sense: there *is* a point on the boundary of the disc where the talyor series diverges: that is what the radius of convergence actually means.
So, analytic functions are gotten by patching information on open connected domains together.

But in some cases, the taylor series on the point that is on the boundary of the disc will not diverge.

Will there be any inconsistencies in complex analysis if the domains were not restricted to be open only (just like in real analysis where domains can be open or closed).

 

[shmoe]

The strength of the complex derivative comes from the limit considering "all directions at once" (compare with what can happen in the real case where we are approaching from 2 directions only). If you drop the "open" requirement, you would include things like the real line, where nice things just don't happen.

There would be no inconsistincies in changing this definition, modulo the fact that you'd have to go around and insert the word "open" before the word "domain" everywhere it appears. It's largely a matter of convenience- open and connected are the properties that we most often require, so this gets a special name.

 

[HallsofIvy]

The basic concept in Topology is the open set: a "topology" for a given set,X, is defined as a collection of subsets of X such that:
1. X is in the collection
2. The empty set is in the collection
3. The union of any collection of sets in the "topology" is also in the "topology".
4. The intersection of any finite number of sets in the topology is also in the topology.

Sets in that topology are the "open sets". I suspect that is why the emphasis is on "open" sets rather than "closed" sets.

 

[Hurkyl]

I certainly wouldn't call it the basic concept -- topologies can be axiomatized in many different ways. A nonexhaustive list would be by the properties of closed sets, by the closure opreator, and even by the collection of "open covers" of your space! And in different contexts, each of these could be more important than the others.

The prime utility of open sets is that they capture the notion of locality. If P is in an open set, then, intuitively speaking, so is everything near P.