Path Components: Examples of Open and Closed?

[pivoxa15]

Homework Statement

Give examples to show Path components need not be open or closed.

I assume it means give one that is not open and one that is not closed.

So one a path component that is open (to satisfy not closed) and one that is closed (to satisfies not open). Or is the question asking for an example that is neither open nor closed?

 

[matt grime]

No, it means give one that is not open or closed. In the example you gave, the path components are open or closed.

 

[pivoxa15]

So its neither open nor closed. So something like (a,b]?

Or (-infinity,0] but is it a path component?

 

[matt grime]

A path component of what? Of what space, and what topology on that space.

 

[pivoxa15]

The question just says any topological space X.

So the question is give examples to show that Path components need not be open or closed in X.

 

[matt grime]

Yes. That was, and still is, the question.

 

[pivoxa15]

So I have to think of a specific topological space X. On R with the usual topology, how about (-infinity, infinity)? It is neither open nor closed? But how does it relate to a path component?

What do you mean by "A path component of what?"? Do you mean I need to specify a specific point?

 

[matt grime]

By (-infinity,infinty) you mean R. This set is both open and closed, as a subset of R (with any topology on R). It is the unique path component of R with its normal topology. But it doesn't help you.

[0,1) is certainly a set that is neither open nor closed in R. But unless you find a topological space of which it is a subset and in which it is still neither open nor closed, and is a path component, then it is useless for the question at hand.

You can't just write down sets, you have to find a topological space. That is why I asked you 'is a path component of what'.

I don't think you understand what a path component is. What is it?

 

[pivoxa15]

Path component of a topological space X are its maximal path connected subsets. So R wouldn't work well to find a path component that is neither open nor closed because they are all open.

 

[matt grime]

There is only one path component of R. Why do you say 'they are all open'? It is things like that that make me think you don't understand what a path component is.

 

[pivoxa15]

So in R the path component is (-infinity,infinty) which is open and closed so not what I am after.

How about consider a path component of R^2 with (0,infinity) x (0,infinity) or the upper right of the plane, not including (0,0). It is not closed because it doesn't contain all its adherent points, namely it doesn't include (0,0). It is not open because it contains more than its interior points, i.e it includes the positive x and y axis.