Why Are Only Constant Functions Continuous in Concrete Topology?

[LeonhardEuler]

Hello, I'm reading this book and I've come to a question that has me stumped:

Show that all functions f:X$\rightarrow$X are continuous in the discrete topology and that the only continuous functions in the concrete topology are the constant functions.

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?

 

[AKG]

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

 

[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

 

[AKG]

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

Then you obviously didn't read the question.

 

[LeonhardEuler]

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

The book says:

In particular, there are always the discrete topology for which T=P(X) and the concrete topology for which T={(empty set),X}.

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.

 

[SpongeBobRhombusHat]

Then you obviously didn't read the question.

No, I read it.

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

 

[DeadWolfe]

It is true that in the discrete topology, the only continuous functions are constant.

How the hell is that?

 

[AKG]

No, I read it.

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

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.

 

[SpongeBobRhombusHat]

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

SBRH

 

[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.

 

[LeonhardEuler]

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

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.

 

[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"?

 

[LeonhardEuler]

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"?

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.

 

[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:

A topological structure is defined to be enough information to decide whether a transformation is continuous or not [18]. The classical definition of continuity depends upon the idea that every open set in the range must have an inverse image in the domain.

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

 

[LeonhardEuler]

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...

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:

Recall that with respect to a discrete topology all maps from the initial to final
state are continuous, while relative to the concrete topology, only the constant functions
are continuous [2]

 

[matt grime]

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:

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

 

[LeonhardEuler]

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

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

 

[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?

 

[LeonhardEuler]

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