Is the convergence in "the topology of pointwise convergence" for sequences or nets? 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!
 A: 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 $x \in X$:

*

*$x_\iota \to x$ in $(X,\mathcal{T})$

*$\forall \kappa \in K: f_\kappa(x_\iota) \to f_\kappa(x)$ in $(X_\kappa,\mathcal{T}_\kappa)$

A: 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.
