The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement

Let $(X,d)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric).

A subset $U \subseteq X$ is open in $(X,d)$ if for every $x \in U$ there is a real $r > 0$ with $B(x,r) \subseteq U$ (Open ball, closed ball and sphere in a metric space). A subset $F \subseteq X$ is closed in $(X,d)$ if its complement $X \setminus F$ is open.

The collection

$\mathcal{T}_d := \{\, U \subseteq X : U \text{ is open in } (X,d) \,\}$

of all open subsets is the metric topology of $d$ on $X$. A subset of $X$ that is both open and closed is called clopen.

Two sets are open for trivial reasons. $\emptyset$ is open, because the defining condition quantifies over no points; and $X$ is open, because $B(x,r) \subseteq X$ for every $x$ and every $r > 0$. Consequently $X$ and $\emptyset$ are also closed, and both are clopen.

A neighbourhood of a point $x$ is any open set containing $x$. The condition above therefore reads: $U$ is open exactly when every point of $U$ has a ball around it inside $U$, and it is the balls alone that have to be tested.

The metric, not the set, determines $\mathcal{T}_d$. Two metrics on the same set may have different metric topologies, and two different metrics may have the same one; the systematic comparison is Topologically, uniformly and Lipschitz equivalent metrics on a set.

• What "topology" means here. $\mathcal{T}_d$ is defined above as a collection of subsets of $X$; the abstract notion of a topological space, a collection of subsets closed under arbitrary unions and finite intersections taken as primitive data, is introduced on a later page and is not used here. What is proved here is that $\mathcal{T}_d$ has exactly those closure properties (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed), which is what licenses the word.
• Open and closed are not opposites. A set may be neither ($[0,1)$ inside $\mathbb{R}$, once the usual metric is available from The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded) or both ($\emptyset$ and $X$ always, and in some spaces every subset at once, as the discrete metric on the companion page shows). "Not open" is never a synonym for "closed".
• Closedness is complementation, and nothing else, at this stage. The description of closed sets by limits of sequences, and the description of the closure as an infimum of distances, are theorems proved later on this page (The closure of a nonempty $A$ is $\{x : d(x,A) = 0\}$, equals $A$ together with its limit points, and is the smallest closed superset, A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed), not part of the definition.