One of the definitions of a subbasis ##\mathcal{S}## of a set ##X## is that it covers ##X##. Then the collection of all unions of finite intersections of elements of ##\mathcal{S}## make up a topology ##\mathcal{T}## on ##X##. That means the collection of all finite intersections of elements of ##\mathcal{S}## is a basis ##\mathcal{B}## for the topology ##\mathcal{T}##.(adsbygoogle = window.adsbygoogle || []).push({});

But one of the defining characteristics of a basis is that it also must cover ##X##, although if the subbasis is the collection of all singletons in ##X##, which definitely covers ##X##, then the basis ##\mathcal{B}## would have only the empty set; wouldn't it?

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# Definition of a subbasis of a topology

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