Reading Munkres' text on Topology, we get the fairly straight-forward definition of a basis:
$\mathcal{B}$ is a basis for a topology on $X$ if $\mathcal{B}$ is a collection of subsets of $X$ such that
(1) For each $x\in X$, there is at least one basis element $B$ containing $X$.
(2) If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there us a basis element $B_3$ containing $x$ such that $B_3 \subset B_1 \cap B_2$.
Now, Munkres proceeds to (roughly) define the topology $\tau$ generated by $\mathcal{B}$ contains elements $U$ so that for each $x \in U$ there is a basis element $B \in \mathcal{B}$ such that $x \in B$ and $B \subset U$.
Consider the set $X = \{a,b,c\}$. Then, by definition, $\mathcal{B} = \{\{a\},\{b\},\{c\}\}$ is a basis for a topology on $X$. We proceed to (attempt to) find the topology generated by $\mathcal{B}$. Clearly, $\{a\},\{b\},\{c\} \in \tau$. We note that given our definitions, the topology $\tau$ generated by $\mathcal{B}$ is $\{X, \emptyset, \{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\}$.
Is this correct, or have I misunderstood something? In particular, does this mean that we may have bases of different cardinalities?
Best Answer
You misunderstood something. Since $\tau$ only includes subsets of $X$. It does not include $\mathcal P(X)$ itself as an element.Instead it includes every element of $\mathcal P(X)$ (read: subset of $X$) which can be written as the union of these singletons. Which subsets are these?Yes, this is correct. Every subset of $X$ is open in this case.
As for the final question, yes, it is possible to have bases of different cardinalities, for example by taking $\{\{a\},\{b\},\{c\},\{a,b\}\}$ you obtain another basis for the same topology, and of course the topology itself is always a basis for itself.