Basis of a Topology

Jun 2016
1
0
New York
I am presently reading about the concept of a topological basis, and I seem to be encountering conflicts between various sources. The definition of basis I am working with is the following:

If \(\displaystyle X\) is a set, a basis for a topology on \(\displaystyle X\) is a collection \(\displaystyle \mathcal{B}\) of subsets of \(\displaystyle X\) such that

(1) For each \(\displaystyle x \in X\), there exists a basis element \(\displaystyle B\) such that \(\displaystyle x \in B\)

(2) If \(\displaystyle x \in B_1 \cap B_2\), where \(\displaystyle B_1\) and \(\displaystyle B_2\) are basis elements, then there is a basis element \(\displaystyle B_3\) such that \(\displaystyle x \in B_3 \subseteq B_1 \cap B_2\).

Here is where the confusion begins. Some sources (I can't reference any in particular, because I have looked through too many) simply define the topology \(\displaystyle \tau\) generated from the basis \(\displaystyle \mathcal{B}\) as that set containing every arbitrary union of elements in \(\displaystyle \mathcal{B}\), while some (such as Munkres) define it as \(\displaystyle \tau = \{ \mathcal{O}~|~ \forall x \in \mathcal{O}, \exists B \in \mathcal{B} ~s.t.~ x \in B \}\). Are these definitions equivalent or are they two different notions with a similar name?

I have another question, in regards to the first of the two definitions. If we are only taking unions, how could one obtain the empty set through these unions?
 
Last edited:

Plato

MHF Helper
Aug 2006
22,490
8,653
If \(\displaystyle X\) is a set, a basis for a topology on \(\displaystyle X\) is a collection \(\displaystyle \mathcal{B}\) of subsets of \(\displaystyle X\) such that
(1) For each \(\displaystyle x \in X\), there exists a basis element \(\displaystyle B\) such that \(\displaystyle x \in B\)
(2) If \(\displaystyle x \in B_1 \cap B_2\), where \(\displaystyle B_1\) and \(\displaystyle B_2\) are basis elements, then there is a basis element \(\displaystyle B_3\) such that \(\displaystyle x \in B_3 \subseteq B_1 \cap B_2\).
The bad news is that it is true that there in no one set of definitions. But the good news it that Almost all sets definitions produce the same results.

As for your concern that there seems to be no way to get the empty set, there is a simple and many say unsatisfactory way from set theory: the union of an empty collection equals the empty set.