Lower limit topology, also known as the standard topology on the real numbers, is a specific topology defined on the set of real numbers \(\mathbb{R}\). This topology is generated by a basis consisting of all half-open intervals of the form \([a, b)\) where \(a < b\).
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The lower limit topology on has basis of a topology . Every such nonempty interval contains a rational, so this topological space is separable. Its topology is finer than the usual Euclidean topology; the one-sided basic neighbourhoods allow its square to have an uncountable discrete antidiagonal.