1. May 15, 2006

### LeonhardEuler

Hello, I'm reading this book and I've come to a question that has me stumped:
I got the first part: since every set in the discrete topology is open, then the inverse image of every open set must be open.

Its the second part that's giving me trouble. It seems to me like many functions could be continuous in the concrete topology even if they are not constant. For example, the identity function: The only open sets in the concrete toplogy are the empty set and X. The inverse image of the empty set is the empty set and the inverse image of X is X. So the inverse images of all the open sets are open. I can't seem to find any flaw in this reasoning. Can someone help me out?

2. May 15, 2006

### AKG

Assuming "concrete topology" does in fact mean that only the full and empty sets are open, your reasoning has no flaw.

3. May 15, 2006

### SpongeBobRhombusHat

Since the question about concrete topology is a follow up to a question about discrete topology, I'd bet that the question was intended to have concrete replaced with discrete.

SBRH

4. May 15, 2006

### AKG

Then you obviously didn't read the question.

5. May 15, 2006

### LeonhardEuler

The book says:
So I don't think I'm misinterpreting that. I guess I just won't get hung up on this as long as I know I'm not crazy. Thank you.

Last edited: May 15, 2006
6. May 15, 2006

### SpongeBobRhombusHat

It is true that in the discrete topology, the only continuous functions are constant. This is not true with the concrete topology. If X is more than one point, then f(x)=x for all x in X is a continuous, non-constant function in the concrete topology.

SBRH

7. May 15, 2006

How the hell is that?

8. May 15, 2006

### AKG

Are you sure you read the question? Just before it asks about the concrete topology, it asks to prove that EVERY function X -> X in the discrete topology is continuous. Why would the next question be to show that only constant functions are continuous in the discrete topology? And you're wrong, in the discrete topology it isn't only the constant mappings that are continuous. All functions are. The original poster proved it in the first post.

9. May 16, 2006

### SpongeBobRhombusHat

Wow! I am being totally dense. Sorry. It was one of those days.

SBRH

10. May 17, 2006

### Jimmy Snyder

What is the name and author of the book? What page#?

Perhaps the author refers to the fact that the only continuous maps from the concrete topology to the discrete topology are the constants.

11. May 17, 2006

### LeonhardEuler

The book is "Tensor Analysis on Manifolds" by Richard Bishop and Samuel Goldberg. The problem was on page 13 and the quote was from page 8.

12. May 17, 2006

### Jimmy Snyder

Thanks LeonhardEuler, I will attempt to look this up when I get a chance. In the meantime, is there any possibility that the authors meant "from concrete to discrete"?

13. May 17, 2006

### LeonhardEuler

I just re-checked and the wording is exactly like I have it. It is possible that the author meant that and didn't write it. The copy I have is the first printing of the book.

14. May 17, 2006

### matt grime

There is certainly a mistake somewhere. Interestingly a google search reveals the following paper citing this result with this reference:

http://132.236.180.11/pdf/math-ph/0101032

the article is pdf and seemingly complete nonsense (heat is a one-form??)

It also contains links to other articles that are amusing in some sense. One of them (the one cowritten with P Bawldin asserts that:

Well, every set in the range has an inverse image in the domain by definition of domain....

Last edited by a moderator: Apr 22, 2017
15. May 17, 2006

### LeonhardEuler

That's really weird: the one thing the author of the first article cites from the book is the very mistake this thread is about:

Last edited by a moderator: Apr 22, 2017
16. May 17, 2006

### matt grime

That's sort of why I posted it.....

17. May 17, 2006

### LeonhardEuler

Oh, I see that now. I didn't take in the full meaning of your first sentence when I first read it.

18. May 17, 2006

### Jimmy Snyder

As has already been noted, the paper by Kiehn (linked to by Matt) cites the very problem that got this thread started. Intriguingly, the actual citation (footnote #2) is to page 199 of the book by Bishop and Goldberg. LeonhardEuler, is this also an error, or is there something on page 199 related to this matter?

19. May 17, 2006

### LeonhardEuler

There is nothing about discrete or concrete topology on page 199. It's mostly about Stoke's Theorem.