Lower limit topology
= Lower limit topology
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= Sorgenfrey line
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The lower limit topology on $\mathbb R$ has <basis of a topology> $\{[a,b):a<b\}$. 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.