Why can't a base of a topology be {X}?

Question

Let $X$ be a non-empty set with topology $T$. Why can't a base of $T$ be $\{X\}$ ? It seems to satisfy the two conditions.

Base of a topology

The answer in this link implies that it can't. I would think that $\{\{a,b\}\}$ is a base as well.

It's also making one of my proofs difficult to do {given that it is a base).

It can: that’s a base for the indiscrete topology ${X,\varnothing}$ on $X$. — Brian M. Scott, Aug 25 '16 at 03:09
http://math.stackexchange.com/questions/241210/base-of-a-topology the answer in this link says otherwise — user40300, Aug 25 '16 at 03:10
@user40300: Can you be more specific? I don't see anywhere on that post that anyone said ${X}$ couldn't be a base. — Eric Wofsey, Aug 25 '16 at 03:12
@BrianM.Scott; a topology $\tau$ is formed from a base $\mathcal B$ by an arbitrary union of the members of the base $\mathcal B$ ;How can we generate the set $\emptyset $ here? — Learnmore, Aug 25 '16 at 03:15
@BobWilson: See http://math.stackexchange.com/questions/1826226/why-is-this-the-basis-for-the-indiscrete-topology/1826230#1826230 — Eric Wofsey, Aug 25 '16 at 03:16
@Bob: $\varnothing$ is the union of the empty subset of ${X}$. — Brian M. Scott, Aug 25 '16 at 03:21
So i believe u guys now, thanks. I'm gonna have to ask another question then -_- — user40300, Aug 25 '16 at 03:25

score 2 · Answer 1 · answered Aug 25 '16 at 03:19

Actually, $\{X\}$ can be a base for a topology (namely, the indiscrete topology, whose only open sets are $X$ and $\emptyset$). In the answer you linked, Brian Scott was listing all the possible bases for a particular topology, namely the topology $\big\{\varnothing,\{a\},\{b\},\{a,b\}\big\}$ on the set $\{a,b\}$. Note that $\{\{a,b\}\}$ is a base for some topology on $\{a,b\}$, it's just not a base for that particular topology.

Why can't a base of a topology be {X}?

1 Answers1