# Continuity of countable projection from non-first countable topological space

This might be trivial, but I just want to make sure I got this right:

Let $$X$$ be a metric space and $$I$$ an uncountable index set. Let us consider $$X^I$$ with the product topology (of course, the topology of $$X$$ is the one induced by the metric). Let $$J \subseteq I$$ be an (infinite) countable subset of $$I$$. Let us also consider $$X^J$$ with the product topology and consider the map

$$F: X^I \to X^J$$ given by $$F((x_i)_{i \in I}) := (x_j)_{j \in J}$$, i.e. $$F$$ is simply the projection from $$I$$ to $$J$$.

Note that $$X^I$$ is not a first countable topological space, so continuity of $$F$$ is not the same as sequential continuity of $$F$$. Now my question: Is the map $$F$$ continuous? It is trivially sequentially continuous and if $$X^I$$ was first countable, then the statement follows. However, I am not used to working with nets instead of sequences (which is necessary to check continuity of $$F$$, since its domain is not first countable) - so I wanted to make sure that my intuition that $$F$$ is continuous is correct. Intuitively, nothing can go wrong, because when I want to check continuity via nets, since I consider the same topology on both spaces, it seems to be straightforward.

Thankful for any answers on this!

Of course this map is continuous: if $$\pi_i$$ is the projection onto the space $$X_i$$ then any map $$F$$ into a product is continuous iff all compositions with all $$\pi_i$$ ($$i$$ in its index set), i.e. $$\pi_i \circ F$$ are continuous. This is the continuity characterisation of all initial topologies (e.g. see my answer here). And for your $$F$$ we just have $$\pi_i \circ F = \pi_i$$ for all $$i \in J$$ trivially, so $$F$$ is continuous.