General Question

lapilofu's avatar

If f:X->Y is continuous and x is a limit point of subset A of X, is f(x) a limit point of f(A)?

Asked by lapilofu (4320points) February 27th, 2010

I heard a rumor that this is not true, but I can’t come up with a counterexample or a proof in favor of it.

Observing members: 0 Composing members: 0

15 Answers

rebbel's avatar

And here’s me, thinking i was the only one who heard that rumour…

grumpyfish's avatar

Wolfram|Alpha isn’t sure how to compute an answer from your input

And neither is grumpyfish =)

janbb's avatar

Don’t believe everything you hear.

Pseudonym's avatar

umm, yeah, sure, whatever you say, I completely understand

buckyboy28's avatar

Obviously you have to diagulate the trimanuglas. Everyone knows that.

grumpyfish's avatar

@lapilofu I’m sorry no one who understands topology has come along yet =) I did actually look up some of what you’re talking about, but it’s outside the math I’ve handled.

lloydbird's avatar

For those not familiar with your terminology (myself included)….Can you rephrase your question in a more widely known and simpler way, please?

Jeruba's avatar

I’d suggest employing the Socratic method to induce recollection of the truths you already know.

lapilofu's avatar

@lloydbird I’m afraid the question is really only for people who have some experience with topology or advanced set theory. To rephrase it in layman’s terms would require summarizing a course in set theory.

lloydbird's avatar

@lapilofu Oh, I see.
Then is there a website, that you could direct me to, where I (and whoever else wants to)
can access a summary of a “course in set theory”?
I’d like to be able to respond at some future date.
Rather than not at all.

lapilofu's avatar

@lloydbird Wikipidia’s pages on Set Theory and Topology are a pretty good sources, though there’s no replacement for a good textbook. Munkres’s “Toplogy: A First Course” is what I’ve been using and it’s pretty clear.

You seem offended by my response, and I’m sorry if that’s the case, I just honestly can’t think of how to simplify my question without typing pages of definitions and theorems.

gailcalled's avatar

Where is @finkelitis when you need him?

ratboy's avatar

Let X be the Reals with the usual topology, and let Y be a space with the discrete topology. Let f:X->Y be constant, say f(x) = y for all x in X. Then f is continuous since the preimage of any subset B of Y under f is either X or empty depending on whether y is in B.  Let A = [0,1); then 1 is a limit point of A, while f(1) is not a limit point of f(A) = {y}, since {y} is an open subset of f(A) that contains no member of f(A) distinct from y.

lloydbird's avatar

@lloydbird I am not in the least offended by your response. No apology needed.
Thanks for the tips. :-)

lapilofu's avatar

Thanks, @ratboy! That’s great!

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