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In the past, I've always motivated continuity of a function from (some subset of) $\mathbb R$ to $\mathbb R$ based on the (incomplete) definition $\lim_{x \to c} f(x) = f(c)$; continuity at isolated points was never really too hard to explain. The main tool behind my explanation is a "using your common sense, what should the value here be?" prompt.

My usual example is this: You have a video camera, you're taking a video of the projectile motion of a ball or something. At precisely 3 minutes, your battery dies. Your camera is so awesome that it records stuff up to 2.999...[however many (finitely many) 9's] minutes. Common sense, i.e. "how the real world works", should tell you that this is enough for you to find out where the ball is at 3 minutes, even without any footage at that specific instant. If it were just a little bit higher/lower than "the right answer", then the ball would have to magically teleport from one point to another in [snaps fingers], which doesn't make any sense.

Then I get to the punchline - that continuity is required for us to "predict" things within "reasonable expectations". Things don't magically jump/teleport from one point to another, neither does temperature, brightness, etc. At isolated points, there's not enough "surrounding data" for us to make predictions, so any value can potentially be "reasonable", because if you claim that my prediction is "too high/low", I can always argue that it's conceivable that there might be a spike/dip in the surrounding data, so who's to say my prediction is definitely wrong?

(If this is a bad explanation, please let me know how I can improve it.)

Fast-forward to a rather different scenario, now I'm trying to explain that the product topology is the coarsest topology that makes all projections continuous. I got countered with "but why should all projections be continuous?" I was stumped. I'd never really thought about the intuition behind continuity outside of $\mathbb R$. So my question is, can I generalize this "prediction" thing to motivate why projections of a product space onto its factor spaces should always be continuous? If so, how? If not, why not?

I have read the answers here, but I think/hope my question is a little different.

Also, feel free to yell at me if I've gotten continuity wrong for the last few years (and poisoned the minds of many others). But I would appreciate it if you could follow up with suggestions. Thanks.

Community
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    sidenote- your intuition of continuity is completely correct (i.e. can easily be made into an alternate definition of continuity) for second countable Hausdorff topological spaces. –  Jul 03 '12 at 22:30
  • The issue for a function $f: X \to Y$ between topological spaces is: what is the meaning of $\lim_{x \to c} f(x)$? One may use the notion of "net" to formulate the definition and then, with that definition, the condition you state is equivalent to continuity at $c$, provided you state it a bit more precisely: for every net $(x_i){ \in I}$ that converges to $c$ in $X$, the net $\bigl((f(x_i)\bigr){i \in I}$ converges to $f(c)$ in $Y$. – murray Jul 18 '15 at 19:08
  • As to your students' question as to why the projections should be continuous: Is that not a natural generalization of the situation the projections from the cartesian plane onto each of the coordinate axes? – murray Jul 18 '15 at 19:10

3 Answers3

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It’s essentially the same intuition: you want to define the topology on $X\times Y$ in such a way that as you move around continuously in $X\times Y$, your ‘shadows’ on the $X$ and $Y$ ‘walls’ move continuously as well. More specifically, you want to know that as you approach a point $\langle x,y\rangle\in X\times Y$, your $X$ and $Y$ projections approach $x$ and $y$ respectively. Of course you may have some trouble getting across the idea that approach depends on the topology chosen; the connection is really intuitive only for metric and linearly ordered spaces. Perhaps it’s better to avoid the idea of approaching a point dynamically and look at limit points. You want to choose the topology of $X\times Y$ so that if $\langle x,y\rangle$ is so close to $A$ that it can’t be separated from $A$ by an open set $-$ i.e., if it’s in $\operatorname{cl}_{X\times Y}A$ $-$ then $x$ and $y$ are so close to the ‘shadows’ $\pi_X[A]$ and $\pi_Y[A]$, respectively, that they can’t be separated from them.

Brian M. Scott
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As countinghaus pointed out in the comments, your intuition for continuity is correct as long as the topological space is second countable and Hausdorff. The intuition to which I am referring is the condition $\lim_{x\to c} f(x)=f(c)$ for all $c\in X$.

However, for general topological spaces, we encounter problems with this intuition. If we remove the condition that the space be Hausdorff, then the problem is that the question: "using your common sense, what should the value here be?" may have multiple (even infinitely many) correct answers (where correct means "makes the function continuous"). For example, if we equip a set $Y$ with the trivial topology, then every function $f:X\to Y$ is continuous (here $X$ denotes an arbitrary topological space). Suppose we look at a function $f:X\to Y$ where $f(x)=c$ for all $x\not=x'$ with $c\in Y$, $x'\in X$. Then we can ask the question "using your common sense, what should the value at zero be?", we might expect $f(x')=c$ to be the correct answer (it makes $f$ a constant function), but this is not a unique answer because any value for $x'$ makes $f$ continuous!

The necessity of the second countability condition is more subtle. Let $X$ denote the long line. If you are unfamiliar with this space, I recommend reading about it because it is great for finding counterexamples like this one! The idea behind the long line is to "paste" together uncountably many copies of the half-open interval $[0,1)$. Note that the real line $\mathbb{R}$ is homeomorphic to a countable such pasting. The term "long line" comes from the idea that we are simply pasting more of these same intervals, so the resulting space should be "longer" than the standard real line.

Now, consider the function $f:X\to\mathbb{R}$ where $f$ is $\sin(2\pi x)$ on each $[0,1)$ interval. I agree that I did not rigorously define this function but that is because I chose not to rigorously define the long line. Suffice it to say that it is not hard to rigorously define the function $f$, but the intuition is clear. Now, We might expect that $f$ is continuous (for if we did this with a countable pasting, we would obtain the function $\sin(2\pi x)$ on $\mathbb{R}$, which is continuous). However, it can be shown that every continuous function whose domain is the long line is eventually constant. That is, for intervals far enough down the line, the function maintains a constant value. Since our function $f$ is not eventually constant, it cannot be continuous.

This is why we need our topological space to be second countable and Hausdorff in order for your intuition to be the correct one. As for your question on the product topology specifically, I believe Brian Scott has handled this superbly in his answer.

J. Loreaux
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For the product, you can take the categorical view either expressly or implicitly.

To take it expressly, note that a product is both an object (the product of sets) and projection morphisms. We need the projections to be in our category, which means we need them to be continuous.

To take it implicitly, you should relate the product to the components: an element of the product is completely and uniquely determined by its components/images under the projections. In other words: the product is really a way of "knitting together" information about what happens with both factors. So a pair of maps into both factors can be "coded" as a map into the product, and a map into the product can be "decoded" as a pair of maps into the factors. This will require the projections to be continuous, as can be seen by considering the map into the product induced by the identity on one factor and constant functions on the other.

Arturo Magidin
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