Examples of Separable Spaces that are not Second-Countable

Question

In this post I give an example of a separable space that is not second-countable. What are other good examples?

Answer 1

Your example is called the Niemytzki plane. It's indeed separable, Tychonoff, non-normal, and it has no countable base because of an uncountable discrete subspace.

Another classic is the Sorgenfrey line (or lower limit topology), which is the reals in the topology whose base consists of all sets of the form $\{[a,b): a < b, a,b \in \mathbb{R}\}$.

One can show that this space is hereditarily separable (all subspaces are separable, e.g. the rationals are dense in the whole space), hereditarily Lindelöf, but does not have a countable base. To see the latter, suppose $B_i, i \in I$ is a base for the Sorgenfrey line. Then for every $x \in \mathbb{R}$, there is some $B_{i(x)}$ such that $x \in B_{i(x)} \subseteq [x, x+1)$. Then if $x \neq y$, say $x < y$, then $x \notin B_{i(y)}, x \in B_{i(x)}$, so $B_{i(x)} \neq B_{i(y)}$ which shows that there are at least as many different base elements as there are points of $\mathbb{R}$.

A closely related ordered space example is the Double Arrow: $[0,1] \times \{0,1\}$, in the topology induced by the lexicographic order: $(x,i) < (y,j)$ iff $x < y$ or $x = y, i = 0, j = 1$. This is a compact ordered (so hereditarily normal) space which is separable (rationals in both parts) and whose subspace $(0,1) \times \{1\}$ is just homeomorphic to the Sorgenfrey line (so it too cannot be second-countable).

If you don't care about separation axioms: an uncountable set in the cofinite topology (a set is closed iff it is finite or $X$) has the property that every countable subset is dense, but there is no countable base.

Or, the included point topology with respect to $0$ on the real line: a set is open iff it is empty or contains $0$. Then $\{0\}$ is dense by itself, but a base should contain all sets $\{x,0\}$ for $x \in \mathbb{R}$, so is always uncountable.

More advanced examples are $[0,1]^I$ in the product topology for an uncountable set $I$ of size $\le |\mathbb{R}|$, or the Cech-Stone compactification of the natural numbers $\beta \mathbb{N}$. A countable dense subset of the product above is even a countable normal space, that is not first countable at any point, which certainly implies that it has no countable base.

More examples can be found at $\pi$-base, an online database of topology examples.

Answer 2

$\pi$-Base is an online encyclopedia of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It currently lists twenty-one separable spaces that are not second countable. You can learn more about any of these spaces by viewing the search result.

Appert Space

Arens-Fort Space

Deleted Diameter Topology

Deleted Radius Topology

Finite Complement Topology on an Uncountable Space

Half-Disc Topology

Helly Space

Maximal Compact Topology

Niemytzki's Tangent Disc Topology

Novak Space

One Point Compactification of the Rationals

Pointed Rational Extension of the Reals

Rational Extension in the Plane

Rational Sequence Topology

Right Half-Open Interval Topology

Single Ultrafilter Topology

Sorgenfrey's Half-Open Square Topology

Stone-Cech Compactification of the Integers

Strong Parallel Line Topology

Strong Ultrafilter Topology

Uncountable Particular Point Topology