Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Here is the reasoning I think fleshes things out more. Some say that the fact follows obviously from the definitions (which may be true), but students often want an answer that explains in some level of detail.

What do you think of the following answer? Does this explain enough or are there areas that need more elaboration. Any comments are appreciated.

share|improve this question
add comment

1 Answer

Let $f: M \rightarrow N $ where M and N are metric spaces (or Topological spaces)

By the topological defintion of continuity, $f$ is continuous iff the pre-image of each open(closed) set in N is also open(closed) in M.

Now, pick any open subset $X \in N$, the pre-image of $N$ under $f$ must be open since we are given that every subset of $M$ is open (since it is clopen). QED.

One can run the same argument with closed sets.

share|improve this answer
Isn't that the argument one has in mind when one says "folows obviously from the definition"? And you may want to replace metric with more general topological spaces. –  Hagen von Eitzen Oct 25 '12 at 18:23
@HagenvonEitzen: yes, but I have had plenty of my friends come to me and ask why and how. For some, it may be difficult to see how it follows from definition chasing; for others, it is natural. Regardless, mathematics should be made clear whenever possible. I just thought it would be nice to just flesh things out. I generally don't like to "keep in the head" and say "obviously" if I have a chance to just say it. if you have the argument in mind, then either say it or explain briefly how that happens. Thanks, I will make the proposed edit. –  user43901 Oct 25 '12 at 18:27
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.