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TLDR: I'm looking for an explicit map that is an open map but not continuous.

The context my question arose was when learning the topological definition of continuous function. I made some progress thanks to this question I wrote. However, there still seems to be a crucial point that I miss conceptually that I believe/suspect will be very useful, which is distinguish clearly open maps and continuous maps. In search for this I found this question:

Open and Closed mapping Examples

which provides a non-constructive argument. But I really wanted to have a constructive argument (especially if the example was from a function from $\mathbb R$ to $\mathbb R$) of a function that is open but not continuous (I'm not interested in closed maps).

Does someone know how to construct one explicit example of it? Or at least the very least explain to me why its so hard to construct an explicit example in this case? (though that would not satisfy me as much).

I just find it unbelievable that its so hard to construct one since there has been put so much emphasis to me that they are not the same. If they are not the same then why can't we construct a simple example that just makes this fact obvious? Thats why I'm looking for hopefully, a instructive example in the simplest spaces I could think of, $\mathbb R$.

One of the main points I hope to get out of this is to understand why is it that continuous functions are able to keep nearby by points nearby while open functions do not? This is crucial for me conceptually.

Charlie Parker
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  • An explicit example of what, exactly? – Lee Mosher Jul 09 '18 at 18:19
  • @LeeMosher sorry if that was unclear, I will add details/adjust title. I am looking for a function (hopefully from R to R or at least simple to understand) that has the property "an example of a function that is open but not continuous". My question came up due to this other question: https://math.stackexchange.com/questions/2052805/open-and-closed-mapping-examples?rq=1 – Charlie Parker Jul 09 '18 at 18:58
  • See https://math.stackexchange.com/questions/2748679/open-but-not-continuous – Bib-lost Jul 09 '18 at 19:19
  • @Bib-lost looks promising! I will take a look at it. I hope that it helps me understand why open maps are not the definition of continuity, which is what I'm really after. – Charlie Parker Jul 09 '18 at 19:33
  • Well, it is easy to see that an enormous amount of very continuous functions will not be open (e.g. the sine function, polynomials of even degree, constant functions, ...) – Bib-lost Jul 09 '18 at 21:17
  • Any function with an isolated local maximum or minimum will not be open. This also covers the sine function, $f(x)=x^2$ etc. – Henno Brandsma Jul 10 '18 at 13:35
  • @HennoBrandsma if the sets are not open I am not interested. I am only looking for open maps that are not continuous. Sorry, I've tried clarifying my question more in my main body, hope it helps. – Charlie Parker Jul 10 '18 at 15:35
  • An open map f will map a nbhd $U$ of $x$ to a nbhd $V$ of $f(x)$ but there is no reason that $V$ must be small just because $U$ is small. – DanielWainfleet Jul 10 '18 at 17:57
  • My point was, there are (lots of) functions from the reals to itself that are open and not continuous. But by necessity they will all be rather "wild" and most of them I think will be quite non-constructive, like the Darboux functions or those built on the Vitali-equivalence (as in other answers on this site). – Henno Brandsma Jul 10 '18 at 21:48

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This may not be what you are looking for, but here goes: There exists a function $f:\mathbb R\to \mathbb R$ such that $f(I)=\mathbb R$ for every interval $I$ of positive length. Such a map is clearly open, but is nowhere continuous. I build such an $f$ here: Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$?

zhw.
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I'm not sure if this is what you want, but here it goes: every map from $\mathbb R$ into itself whose range is finite is a closed map (because every finite subset of $\mathbb R$ is closed), but such a map is continuous if and only if it is constant.

José Carlos Santos
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  • I'm not sure either if thats what I was looking for. I was looking for something that was an open map. So mapping open sets to open sets while at the same time showing its not continuous (since for me intuitively open maps don't seem any different to continuous functions in any obvious way...of course I know I'm wrong but thats why my question exists). – Charlie Parker Jul 09 '18 at 19:05
  • in your example $f(X)$ does not seem to be an open set, so I am not sure how it's related to what I'm looking for actually...hmmm.... – Charlie Parker Jul 09 '18 at 19:16
  • @Pinocchio In your question, you mentioned closed mappings. I explaind how to construct closed mappings which aren't continuous. – José Carlos Santos Jul 09 '18 at 21:09
  • I'm sorry if my question was confusing. I tried clarifying. The question I link mentions closed sets but I'm not actually interested in those. Thanks though. – Charlie Parker Jul 10 '18 at 15:33