Initial topology ($\sigma (f_i:i\in I)$)

For a family of functions $f_i:X\to Y_i$ into topological spaces, the initial topology on $X$ is the coarsest topology making them all continuous. Its subbasis consists of $f_i^{-1}(U)$ for open subsets $U\subseteq Y_i$. A map into $X$ is continuous exactly when all its composites with the $f_i$ are continuous.

On the real plane, take the initial topology of all separately continuous real-valued functions. Its coordinate lines have their usual induced topology, and the nonzero rational grid is dense, but its diagonal is uncountable and discrete. Thus the space is separable and nonmetrizable. Rectangular bump functions show that no sequence from the nonzero rational grid converges to the origin, so the space is not first countable either.