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I am reading about point-set topology. Is empty set always part of a basis of a topology?

hereitis
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    Actually, never. – Moishe Kohan Jul 05 '17 at 04:10
  • Not by the usual definition of basis. – Angina Seng Jul 05 '17 at 04:13
  • I think it could be but in practice it is totally unimportant whether it is or not. – MJD Jul 05 '17 at 04:13
  • @MoisheCohen Huh? What prevents the emptyset from being in a base? There's no minimality requirement ... – Noah Schweber Jul 05 '17 at 04:15
  • Aren't bases closed under intersection? If so, the empty set would be part of the standard base of , e.g., the Reals, by intersecting, say, $(1,0)$ and $(2,3)$. – MSIS Jul 05 '17 at 04:49
  • @MSIS Late response, but: not quite - a base satisfies the weaker condition that given any finitely many elements $B_1,...,B_n$ of the base, then for each $x\in B_1\cap...\cap B_n$ there is a base element $B_x$ with $x\in B_x\subseteq B_1\cap...\cap B_n$. Being closed under finite intersections is sufficient, but not necessary, for this property to hold, and note that if $B_1\cap...\cap B_n=\emptyset$ then there are no relevant $x$ and so this property is trivially satisfied (for that choice of basic opens). This, at least, is the definition in Munkres (and wikipedia and elsewhere). (cont'd) – Noah Schweber Jan 21 '19 at 17:47
  • Annoyingly, I have seen bases defined to be closed under finite intersections as well; however, my understanding is that that is the less common definition (although not being a topologist I could be wrong). – Noah Schweber Jan 21 '19 at 17:47

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It can be, but it need not be: a base for a topological space $(X, \tau)$ is a family of open sets $\mathfrak{B}\subseteq\tau$ such that each $U\in\tau$ can be written as a union of elements of $\mathfrak{B}$. For instance, $\tau$ itself is a base (and $\tau$ certainly contains the empty set). However, it's also true that if $\mathfrak{B}$ is any base, then so is $\mathfrak{B}\setminus\{\emptyset\}$, that is, the set of nonempty elements of $\mathfrak{B}$ - this is because the empty set is the union of an empty collection of sets, so is a union of elements of any collection of sets.

So it can be, but it can also always be safely omitted.

Noah Schweber
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The important property of a basis is that open sets are exactly those sets that are unions of basis elements.

Including the empty set in the basis, or not, does not affect which sets are unions of basis elements, so it doesn't make any difference whether the basis includes the empty set.

(Note that the empty set is always a union of basis elements, since it is the union of no sets.)

MJD
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  • thank you!! I just wonder, given that the empty set is always in the topology, if empty set is not in the basis, will the empty set be in the set of the unions of the basis elements? – hereitis Jul 05 '17 at 04:30
  • now i understand! Thank you! – hereitis Jul 05 '17 at 04:31