4

Actually, this is only a clarification about the definition of a base for a topology. In the book of Dshalalow entitled "Real Analsysis: An Introduction to the Theory of Real Functions and Integration", CRC Press LLC, USA, 2001, at p.115, defined the base for a topology as follows and I quote:

Definition. Let $(X,\tau)$ be a topological space. A subcollection $\mathcal{B}$ of open sets is a base for $\tau$ if every open set is a union of some elements of $\mathcal{B}$. (Specifically, it follows that $\varnothing$ must be an element of $\mathcal{B}$.)

I got confused with the definition of a base because as Scott mentioned in here, it is never necesssary to include $\varnothing$ in a base. I want to be clarified with this and many thanks in advance.

Community
  • 1
Juniven Acapulco
  • 9,654

1 Answers1

8

Since $\varnothing$ is a subset of any collection $\mathscr{B}$, and $\bigcup\varnothing=\varnothing$, it really isn’t necessary to include $\varnothing$ in $\mathscr{B}$: the empty set is the union of the empty sub-collection of $\mathscr{B}$. However, some people don’t like to deal with (or simply forget about) $\bigcup\varnothing$; that’s probably what happened here.

Note that in practice $\varnothing$ is very often not included in a given base for a topology: very often one describes a local base at each point of the space and then takes the union of these local bases as a base for the topology, and when this is done, $\varnothing$ is not included. Similarly, the family of open intervals $(p,q)$ with $p,q\in\Bbb Q$ and $p<q$ is often given as a base for the Euclidean topology on $\Bbb R$, and it does not include $\varnothing$.

Brian M. Scott
  • 616,228