# $X$, an uncountable set, with the cofinite topology is not first-countable?

Is the following a valid proof that $$X$$, an uncountable set, with the cofinite topology is not first-countable?

By way of contradiction, suppose that $$\mathscr{B}_x=\left\{B_n:n\in\mathbb{N}\right\}$$ is a countable local base of open sets at the point $$x$$.

Since $$X$$ is $$T_1$$ and $$\mathscr{B}_x$$ is a local base at $$x$$,

$$\bigcap_{n\in\mathbb{N}} B_n = \{x\}.$$

By De Morgan,

$$X\setminus\bigcap_{n\in\mathbb{N}} B_n = X\setminus\{x\}$$

$$\bigcup_{n\in\mathbb{N}} X\setminus B_n = X\setminus\{x\}.$$

As $$B_n$$ is open the set $$X\setminus B_n$$ is finite, hence, it is countable. Notice, we have a countable union of countable sets, hence, a countable set. Therefore, it must be that $$X\setminus\{x\}$$ is countable, which implies $$X$$ is countable, a contradiction.

• Looks good to me. – triple_sec Jan 31 at 3:39
• Can you argue why $\bigcap\limits_{n\in\mathbb{N}}B_n = \{x\}$ ? – jMdA Jan 31 at 3:47
• @jMdA: Yes. Are you asking for the proof? It followings from the fact that $X$ is $T_1$. – user853982 Jan 31 at 4:23
• @jMdA. If $x\ne y\in X$ and if $B_x$ is any local base at $x$ then $X\setminus \{y\}$ is a nbhd of $x$ so there exists $b\in B_x$ with $b\subset X\setminus \{y\}.$ So $y\not \in \cap B_x.$ – DanielWainfleet Jan 31 at 12:14

Using $$\bigcap_n B_n = \{x\}$$ is not directly necessary, but is OK in your proof. So your solution is fine as soon as you can justify this intersection from $$T_1$$-ness (some earlier result in your text presumably).
You could also argue: $$\bigcup_n X\setminus B_n$$ is at most countable, so we can pick $$y \in X$$ that is not in this union and unqual to $$x$$, as $$X\setminus \{x\}$$ is uncountable. Then $$U= X\setminus \{y\}$$ is open (definition of cofinite topology, it has finite complement) contains $$x$$ and so must contain some $$B_m$$. But then $$y$$ was chosen to be not in $$X\setminus B_m$$, so $$y \in B_m$$ and we cannot have $$B_m \subseteq X\setminus \{y\}$$, contradiction.