Counter example to first countable, second countable, and separable spaces

Question

Can anyone give a counter example as easy as possible to

1) Every first countable space is a second countable space.

2) Every separable space is a second countable space.

Answer 1

While the Sorgenfrey line (aka lower limit topology) is a great counterexample that happens to work for both of these questions (and many more), I think it's more instructive to find simpler examples.

Hint:

For the first question, look at the discrete topology on a set $X$. It's easy to see that this is first countable. Can we make $X$ large enough so that this topology is not second countable?

For the second question, look at the cofinite topology on an infinite set $X$. Any infinite subset will be dense, so it is separable. Can we make $X$ large enough so that this topology is no longer second countable?

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The answer to both of these questions is yes. To see this, one can consider any uncountable set. Indded, an uncountable discrete space is not second countable, and an uncountable set with the cofinite topology is not even first countable (why?).

An alternative approach (that's worth thinking about only after you understand the above reasons), is the following.

A second countable $T_0$ space has cardinality at most $2^{\aleph_0}$.

Proof: Fix a countable basis $\mathscr{B}$ of a $T_0$ space $X$ and define a function $f:X\to\mathscr{P}(\mathscr{B})$ by $$ f(x) = \{B\in\mathscr{B} \mid x\in B\}. $$ This function is an injection, for if $x,y\in X$ are distinct, then by the $T_0$ property there is $B\in\mathscr{B}$ such that $x\in B$ and $y\not\in B$ (or visa-versa). Thus $f(x)\ne f(y)$. Since $f$ is an injection, we conclude $$|X|\le |\mathscr{P}(\mathscr{B})|=2^{\aleph_0}. \tag*{$\square$} $$

Both of the examples above are $T_0$, along with pretty much every nontrivial topological space. Consequently, taking $X$ to have cardinality larger than $2^{\aleph_0}$ will also answer the two questions above.