Understanding Normal Topology & Examples of Non-Normal Topologies

[pivoxa15]

If a question at the end specifies wrt the normal topology, what does it mean?

What would be an example of a topology that is not normal?

 

[NateTG]

If a question at the end specifies wrt the normal topology, what does it mean?

What would be an example of a topology that is not normal?

http://en.wikipedia.org/wiki/Normal_space

 

[AKG]

Maybe they mean "with respect to the norm topology." If your space is a normed vector space, then the norm defines a topology called the norm topology. Or, by "normal" they might mean "usual" as in the topology you usually use. For example, you know what the usual topology is in Rⁿ. NateTG, I don't think that we ever say that a normal space has "the normal topology." For one, there are usually many topologies that could make a given set a normal space, so we wouldn't say "the normal topology." But moreover, we usually wouldn't say "a normal topology," so his textbook probably doesn't mean that.

 

[matt grime]

I agree with AKG. Normal, here, just means the usual, natural, obvious, or standard topology.

 

[pivoxa15]

The questions I do are beginners level topology but then what is the obvious or standard topology? Does the obvious topology <=> R^n?

 

[StatusX]

What is the set? Maybe you should copy the whole question.

 

[pivoxa15]

The sets in the questions have always been R or R^2. What does the usual topology in R or R^2 mean? What would be an example of a topology on R or R^2 that is not usual?

 

[Hurkyl]

What would be an example of a topology on R or R^2 that is not usual?

The discrete and indiscrete topologies, for example. And because the sets R and R^n are bijective for any n, there is a topology on the set R that is homeomorphic to the usual topology on R^n.

 

[pivoxa15]

Could someone tell me what the usual topology on R and R^2 is exactly? Topology is a very new topic to me and nothing in yet is usual yet.

Does the topology define what kind of subsets exist?

 

[Hurkyl]

Could someone tell me what the usual topology on R and R^2 is exactly? Topology is a very new topic to me and nothing in yet is usual yet.

Does the topology define what kind of subsets exist?

Set theory tells you what subsets exist.

A topology on a set X is a specification of certain subsets as being open.

A topology can be generated by a basis of neighborhoods.

One basis for the usual topology on R is the collection of all open intervals. For R^2, the open disks. For R^3, the open balls.

 

[pivoxa15]

Do all sets in a topology must be open? And when a topology is defined is it the case that we can only work with sets defined in this topology?

So the usual topology on R are all open intervals in the form (-infinity,a) or (b,infinity)? a,b in R. Can any intervals like (-infinity,0] appear? If not than does that mean we can use intervals like that in the question?

Or is it the case that given the topology consisting of open sets, we can always create closed sets via taking the complements of the open sets given in the topology. So we can work with closed sets.

 

[matt grime]

Think of a topology as a method of assigning the adjectives 'open', 'closed', or 'neither' to subsets of some space.

This must be done in a way that is compatible with the rules of 'being a topological space'. A set is open if and only if its complement is closed, and thus it is sufficient to state the rules only for one or other of open and closed.

(-infinty,0] is closed in the usual topology. It is not open in the usual topology. (Some sets in some topologies may be both open and closed.)

Open sets are unions of open intervals, in the usual topology on R. So (0,1)u(3,4) is open.

The rationals, as a subset of R, are neither open nor closed.

 

[pivoxa15]

The discrete and indiscrete topologies, for example. And because the sets R and R^n are bijective for any n, there is a topology on the set R that is homeomorphic to the usual topology on R^n.

Why is the topology which calls all sets of X as open called the discrete topology?

 

[matt grime]

Why not? A discrete set is one made up of 'separate' points. Thus the integers are thought of as discrete. The reals are thought of as a continuum. So a discrete subset of something means each point has an (open) neighbourhood around it not containing any other point. The discrete topology means every set is open, so the space as a whole is a discrete subset of itself.