Is Every Singleton Set Open? (No, Here's When It Is)

The statement “every singleton set is open” is false as a general fact, but true in some important spaces. Here is how to tell which situation you are in.

Counterexample: the real line

Take $\mathbb{R}$ with $d(x, y) = |x - y|$ and the set $\{0\}$. For $\{0\}$ to be open, some interval $(-\varepsilon, \varepsilon)$ with $\varepsilon > 0$ would have to lie inside $\{0\}$. But $(-\varepsilon, \varepsilon)$ always contains $\varepsilon/2 \neq 0$. So $\{0\}$ is not open, and the same argument works for every point of $\mathbb{R}$.

When a singleton is open

In a metric space, $\{x\}$ is open exactly when $x$ is an isolated point:

$$\{x\} \text{ is open} \iff \exists\, r > 0 \text{ such that } B(x, r) = \{x\}.$$

Equivalently, $\inf_{y \ne x} d(x, y) > 0$. Examples where this holds for every point, so that all singletons are open:

• Any finite metric space. Take $r$ to be the smallest distance to another point (full proof).
• $\mathbb{Z}$ with the usual distance. $B(n, \tfrac12) = \{n\}$.
• Any set with the discrete metric, where $d(x,y) = 1$ for $x \neq y$. Here $B(x, 1) = \{x\}$.

A space where every singleton is open has the discrete topology: every subset is open, because it is a union of singletons.

Mixed cases exist too. In $X = \{0\} \cup \{1/n : n \ge 1\}$ with the usual distance, each $\{1/n\}$ is open but $\{0\}$ is not.

Singletons are always closed in a metric space

The opposite property does hold in general. To see that $\{x\}$ is closed, show that its complement is open. If $y \neq x$, put $r = d(x, y) > 0$. Then $B(y, r)$ doesn’t contain $x$, so it lies in the complement. Hence $X \setminus \{x\}$ is open and $\{x\}$ is closed.

(In general topological spaces, singletons are closed exactly when the space is $T_1$. Every metric space is $T_1$, but some non-metric topologies are not.)

Summary of cases

Space	Is $\{x\}$ open?	Is $\{x\}$ closed?
$\mathbb{R}$, $\mathbb{R}^n$ (usual metric)	No	Yes
Finite metric space	Yes	Yes
$\mathbb{Z}$, discrete metric	Yes	Yes
Indiscrete topology on a set with $\ge 2$ points	No	No

Common mistakes

• Thinking “open” means “not closed”. Sets can be both (as in discrete spaces) or neither (such as $[0,1)$ in $\mathbb{R}$).
• Proving openness with a closed ball, or with a radius that hasn’t been shown to be positive.
• Generalising from finite spaces. The finite case works only because a minimum over finitely many positive distances is positive.