# Normal topology?

1. May 25, 2007

### 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?

Last edited: May 25, 2007
2. May 25, 2007

### NateTG

3. May 25, 2007

### 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.

4. May 25, 2007

### matt grime

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

5. May 25, 2007

### pivoxa15

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

6. May 25, 2007

### StatusX

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

7. May 25, 2007

### 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?

8. May 25, 2007

### Hurkyl

Staff Emeritus
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.

9. May 26, 2007

### 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?

10. May 26, 2007

### Hurkyl

Staff Emeritus
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.

11. May 26, 2007

### 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.

Last edited: May 26, 2007
12. May 26, 2007

### 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.

13. May 27, 2007

### pivoxa15

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

14. May 27, 2007

### 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.