Definition: Basis of a topology

Question

I have a question regarding the basis of a topology, or actually 2 different definitions I encountered. Munkres: If $X$ is a set, a basis for a topology on $X$ is a collection $\mathscr{B} \subset P(X)$ such that:

i) For each $x \in X$, there is at least one basis element $B \in \mathscr{B}$ containing $x$.

ii) If x belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3 \subset B_1 \cap B_2$

And then he defines the generated topology as $ \{ U \subset X \ | \ \forall x \in U \ : \exists B\in \mathscr{B} \ \ \text{s.t.} \ \ x \in B \subset U \}$.

So this is basically a way to start with a collection of subsets of $X$ that fulfill those two properties and this collection will always be a basis for one topology on this set $X$.

In my class we started from the other direction, namely given a topological space $(X,\tau)$, $\mathscr{B} \subset P(X)$ is called a basis if every open set $U \in \tau$ can be written as an arbitrary union of basis elements.

On my latest exercise sheet, I have to proof that $\mathscr{B} \subset P(X)$ is the basis of a topology $\tau$ on $X$ (in the sense of the second definition) if and only if the two defining properties from Munkres definition hold. The one direction of this is pretty easy by giving a constructive proof using the generated topology. In the other direction, I have to show that if one of those properties does not hold, there can not be a topology $\tau$ on $X$ such that $\mathscr{B} \subset P(X)$ is a basis of $\tau$. If the first property is violated, then $X$ cannot be written as a union of basis elements so this one is clear. However, in my understanding, I have a counterexample for the second one:

Let $X = \{1,2,3\}$ and $\tau = \{ \emptyset, X\}$ be the trivial topology. Now let $\mathscr{B}$ be such that it contains $B_1 = \{1,2\}$ and $B_2 = \{2,3\}$. To see that $\mathscr{B}$ is a basis, we have to check that we can generate $X$ as the union of basis elements which is clearly the case. However $B_1 \cap B_2 = \{ 2\}$ so property ii) is violated.

Now I obviously got something mixed up at some point but I am really not sure where so could somebody please clarify this for me?

Thanks, Max

Answer 1

Your second definition

given a topological space $(X,\tau)$, $\mathscr{B}\subseteq P(X)$ is called a basis if every open set $U\in\tau$ can be written as an arbitrary union of basis elements.

is incorrect. It should be:

given a topological space $(X,\tau)$, $\mathscr{B}\subseteq \tau$ is called a basis if every open set $U\in\tau$ can be written as an arbitrary union of basis elements.

Do you see the subtle yet major difference?

Note how bad your original definition is. According to it $P(X)$ is a basis for every possible topology.