Example to illustrate the necessity for nets?

Question

Given topological spaces $(X,\tau)$, $(Y,\upsilon)$, we say a function $f:X\rightarrow Y$ is continuous iff $\forall$ $U\in \upsilon$, $f^{-1}(U)\in \tau$.

Equivalently, we can say that $f$ is continuous iff $\forall$ convergent nets $x_i \rightarrow x$ in $X$, $f(x_i) \rightarrow f(x)$ in $Y$.


I understand that we need the generalization of sequences (nets) since not all topological spaces are first-countable.
Is there an example to illustrate the point? That is, is there a construction of a function $f$ s.t. $f$ is not continuous (using the first definition), but $f$ satisfies the second definition when limited to sequences? Just curious.

Answer 1

Let $X$ be an uncountable set endowed with the cocountable topology $\tau$: a nonempty subset is open iff its complement is countable. This is a non-discrete space in which the only convergent sequences are the eventually constant sequences.

It follows that for any topological space $Y$ and any function $f: X \rightarrow Y$, $f$ is sequentially continuous, i.e., if $x_n \rightarrow x$ then $f(x_n) \rightarrow f(x)$. But because $X$ is not discrete, the identity map from $(X,\tau)$ to $(X,\text{discrete})$ is not continuous.