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In topology we have a theorem that in the topology of pointwise convergence $f_n\rightarrow f$ iff ($f_n\rightarrow f$ for every $x$).

What I don't is the differenece between saying that $f_n\rightarrow f$ and saying that ($f_n\rightarrow f$ for every $x$).

Isaac
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    The convergence $f_n \to f$ is defined to mean that $f_n(x) \to f(x)$ for all $x$. – David Kraemer Nov 11 '21 at 21:54
  • @DavidKraemer, https://i.imgur.com/9LLybiN.png – Isaac Nov 11 '21 at 21:59
  • @DavidKraemer this is the definition of pointwise convergence but there are other notions of convergence apart from that. See https://math.stackexchange.com/questions/597765/pointwise-vs-uniform-convergence – Gold Nov 11 '21 at 22:12
  • @Gold I agree, but "topology of pointwise convergence" was specified in the question. – David Kraemer Nov 11 '21 at 22:20
  • @DavidKraemer But it's not defined like that (at least not in my experience). The topology is defined to be the one generated by sets of the form ${f:f(x)\in O}$ for $x\in X$ and $O$ open in $Y,$ and then you prove that convergence is pointwise in this topology. – spaceisdarkgreen Nov 11 '21 at 22:27
  • @spaceisdarkgreen Maybe I'm an out of touch practitioner, lol. I never think of function topology characterized by anything but what convergence means. – David Kraemer Nov 11 '21 at 22:28
  • @DavidKraemer I may well be the one out of touch here (but based on the page OP linked, that seems to be the approach they're taking). – spaceisdarkgreen Nov 11 '21 at 22:30

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In general, for a given topological space, we say a sequence $x_n$ converges to $x$ if for every neighborhood $U$ of $x$ there is an $N$ such that $x_n\in U$ for all $n>N.$

If $\mathcal F$ is some set of functions $X\to Y$, we can in general have a lot of different topologies on $\mathcal F$ and which topology we choose will determine the definition of convergence of a sequence of functions in $\mathcal F.$

So what this is saying is that for this particular topology, we have $f_n$ converges to $f$ if and only if $f_n(x)$ converges to $f(x)$ for every $x\in X.$ Note this must mean that $Y$ is a topological space, so we can talk about the convergence of the sequence $f_n(x)$ in $Y,$ and also suggests that this topology of $Y$ will be relevant to the definition of the "topology of pointwise convergence" (better known as the product topology) on $\mathcal F$. If we choose a different topology for $\mathcal F,$ the resulting conditions for convergence will generally be different.

spaceisdarkgreen
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