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  1. Let $\{f_a, a \in[0, \infty) \}$ be a $[0, \infty)$ -indexed family of functions from $\mathbb R$ to $\mathbb R$.

    If any subsequence of the family of functions $\{f_{a_i}, i \in \mathbb N, a_i \in [0, \infty), a_i < a_{i+1}, a_i \to \infty \text{ as } i\to \infty\}$ converges pointwise as $i \to \infty$, will $\{f_a, a \in[0, \infty) \}$ converge pointwise as $a \to \infty$?

  2. Note that $\{f_a, a \in[0, \infty) \}$ is a net of functions from $\mathbb R$ to $\mathbb R$, while $\{f_{a_i}, i \in \mathbb N, a_i \in [0, \infty), a_i < a_{i+1}, a_i \to \infty \text{ as } i\to \infty\}$ is a sequence of functions from $\mathbb R$ to $\mathbb R$.

    Consider the topology of pointwise convergence on the set of functions from $\mathbb R$ to $\mathbb R$, is my above question same as: are net continuity at $\infty$ and sequential continuity at $\infty$ equivalent?

  3. Note that in a sequential spaces, net continuity and sequential continuity are equivalent.

    So I wonder if the topology of pointwise convergence on the set of functions from $\mathbb R$ to $\mathbb R$ is a sequential topology?

    More generally, is the topology of pointwise convergence on the set of functions from a set to a topological space a sequential topology?

Thanks and regards!

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up vote 1 down vote accepted

Since you’re looking at pointwise convergence, your question is equivalent to asking whether the following two assertions about a function $f:[0,\to)\to\Bbb R$ are equivalent:

  1. $\lim_{x\to\infty}f(x)$ exists.
  2. For each strictly increasing sequence $\langle a_k:k\in\Bbb N\rangle$ of non-negative real numbers such that $\lim_{k\to\infty}a_k=\infty$, $\lim_{k\to\infty}f(a_k)$ exists.

(In other words, my $f(x)$ is your $f_x$, and I’m looking at what your functions are doing to a single fixed point.)

The two statements are equivalent. It’s easy to see that (1) implies (2). Conversely, if (1) is false, then there are strictly increasing sequences $\langle a_k:k\in\Bbb N\rangle$ and $\langle b_k:k\in\Bbb N\rangle$ such that $$\lim_{k\to\infty}a_k=\lim_{k\to\infty}b_k=\infty$$ and $$\lim_{k\to\infty}f(a_k)\ne\lim_{k\to\infty}f(b_k)\;.$$ Now let $c_0=a_0$. Given $c_{2k}$, let $c_{2k+1}=b_n$, where $n$ is minimal such that $b_n>c_{2k}$, and let $c_{2k+2}=a_m$, where $m$ is minimal such that $a_m>c_{2k+1}$. Then $\langle c_k:k\in\Bbb N\rangle$ is strictly increasing, $\lim_{k\to\infty}c_k=\infty$, and $\lim_{k\to\infty}f(c_k)$ does not exist.

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Thanks, Brian! (1) By $\lim_{k\to\infty}f(a_k)=\lim_{k\to\infty}f(b_k)$, do you mean $\lim_{k\to\infty}f(a_k) \neq \lim_{k\to\infty}f(b_k)$ instead? (2) can we cay that the topology of pointwise convergence on the set of functions from $\mathbb R$ to $\mathbb R$ is a sequential topology, and the topology of pointwise convergence on the set of functions from $\mathbb R$ to $\mathbb R$ is a sequential topology? –  Tim Mar 27 '13 at 14:41
    
@Tim: You’re welcome! (1) Yes, that was a rather disastrous typo. (2) (Did you intend to ask the same question twice?) The topology of pointwise convergence of functions from $X$ to $Y$ is just the product topology on $Y^{|X|}$, which, as shown here, is never sequential when $Y$ is non-trivial and $|X|\ge\omega_1$. –  Brian M. Scott Mar 27 '13 at 19:56
    
Thanks! I intended to ask if "the topology of pointwise convergence" on $\mathbb R^{\mathbb R}$ is sequential , and if "the topology of pointwise convergence" on $Y^X$ is sequential. –  Tim Mar 27 '13 at 20:44
    
@Tim: Ah, I guessed right, then. As you can see, it isn’t sequential for $\Bbb R^{\Bbb R}$. –  Brian M. Scott Mar 27 '13 at 20:50
    
I missed what your wrote. $\mathbb R$ is both nontrivial and $\geq \omega_1$. By the way, can a sequential topology be defined to be a topology s.t. net continuity and sequential continuity are equivalent? –  Tim Mar 27 '13 at 23:41
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