= Initial topology of separately continuous functions
{title2=$\mathcal T_{\rm sc}$}
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 topological space>[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.
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