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The topology of pointwise convergence on $Y^X$, where $Y$ is a topological space and $X$ is a set, is defined to be the topology that topologize the pointwise convergence of mappings from $X$ to $Y$.

In the definition, I was wondering if the pointwise convergence here is for all nets of mappings or all sequences of mappings? I am thinking it is the former, but in what I have seen sequences are mentioned all the time in a non-definition context that a sequence converges wrt the topology of pointwise convergence iff the sequence converges pointwise.

Or when specifying the topology of pointwise convergence, one has to also specify whether the convergence is for nets or sequences? If nets or sequences are not specified, which one is the default?

Thanks and regards!

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2 Answers

up vote 2 down vote accepted

If you want to define a unique topology, then you cannot stop at sequences. You should define net convergence, and a net convergence can define a topology (if it satisfies the 4 Kelly conditions etc.).

If you just postulate that all sequences converge in $X^Y$ iff they do pointwise, then this does not define a unique topology.

To see this, let $Y$ be the co-countable topology on $\mathbb{R}$, let $X = \{0,1\}$ for concreteness. Then the pointwise convergence (from nets, or per convention, as the initial topology) is just the product topology of 2 co-countable spaces. A sequence converges in it, iff it is eventually constant (in both coordinates, and so overall). But the discrete topology on $Y \times Y$ (as sets) has the exact same behaviour with respect to convergence of sequences, and so do all topologies that lie inbetween them. So we have many topologies on $Y^X$ that have the behaviour that sequences converge iff they converge pointwise.

So the sequence variant cannot function as the definition. It is a nice property to have, but not enough to define the topology. If you really want to do it via that route, then you have to use nets, there is no escaping that.

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Thanks! In the same spirit, when talking about the topology of uniform convergence and the topology of compact convergence, are the convergences in both cases also for nets? If the convergences are for sequences, they cannot uniquely determine their topologies? –  Tim Feb 25 '13 at 18:53
    
I suspect so. But with uniformities one does not specify just convergence but uniform convergence. And I'm not aware (but this is just my ignorance!) of a Kelly like theory for when a "uniform convergence space" uniquely defines a uniformity (and not just a topology; note that in general a topological space can have many compatible uniformities, just like a metrizable space and metrics) –  Henno Brandsma Feb 25 '13 at 18:59
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The topology $\mathcal{T}$ of pointwise convergence on $Y^X$ is defined as the initial topology with respect to the projections $(\pi_x)_{x \in X}$ where $$Y^x \ni f \mapsto \pi_x(f) := f(x) \qquad (x \in X)$$ Let $(f_{\iota})_{\iota \in I}$ a net in $Y^X$ and $f \in Y^X$. Then $f_\iota \to f$ in $(Y^X,\mathcal{T})$, if and only if, $$\forall x \in X: \underbrace{\pi_x(f_\iota)}_{f_\iota(x)} \to \underbrace{\pi_x(f)}_{f(x)}$$ i.e. if the net is pointwise convergent. This fact can be easily concluded from the following theorem.

Let $X$ a non-empty set and $((X_\kappa,\mathcal{T}_\kappa))_{\kappa \in K}$ a family of topological spaces. Let $f_{\kappa}: X \to X_{\kappa}$ a mapping ($\kappa \in K$) and denote by $\mathcal{T}$ the initital topology with respect to $(f_\kappa)_{\kappa \in K}$. Then the following statements are equivalent for a given net $(x_\iota)_{\iota \in I}$ in $X$ and $x \in X$:

  1. $x_\iota \to x$ in $(X,\mathcal{T})$
  2. $\forall \kappa \in K: f_\kappa(x_\iota) \to f_\kappa(x)$ in $(X_\kappa,\mathcal{T}_\kappa)$
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Thanks! I understand you define the topology of ptwise cvg to be the product topology. My definition (in my first and second paragraph) is different, and is in terms of topologizing a given family of nets or sequences together their respective "limits". My question was if the given family is of nets or of sequences. Although your reply doesn't address my question directly, I guess the answer to my question is nets instead of sequences? –  Tim Feb 25 '13 at 17:34
    
@Tim I'm not sure whether I understand you correctly. What do you mean by "topologozing a given family of nets or seq. together their respective "limits""? You wrote nothing abot families of nets/seq. in your post above at all. Do you want to define a topology $\tau$ on $Y^X$ such that $f_n \to f$ in $(X,\tau)$ iff $f_n \to f$ pointwise ... and asking whether this equivalence then also holds for nets? (By the way, it's not only "my definition" of topology of ptw. cvg. - it's probably the most popular one.) –  saz Feb 25 '13 at 18:32
    
"Do you want to define a topology $τ$ on $Y^X$ such that $f_n→f$ in $(X,τ)$ iff $f_n→f$ pointwise" yes, that is what I meant, except that I am asking whether $f_n$'s are any net or sequence. "asking whether this equivalence then also holds for nets?" Not really, as I mentioned, my question of whether it is about nets or sequences comes before finding the topology on $Y^X$ not after finding one. –  Tim Feb 25 '13 at 18:46
    
@Tim As Henno Brandsma already pointed out it doesn't suffice to ask for pointwise convergence of sequences to define a unique topology. That's why on defines the topology using projections (which leads to pointwise convergence of nets). –  saz Feb 25 '13 at 18:50
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