Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not

A topological space $(X, \mathcal{T})$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) is metrizable if there is a metric $d$ on $X$ (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric) whose metric topology is $\mathcal{T}$, that is $\mathcal{T} = \mathcal{T}_d$ (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). Such a $d$ is said to induce or metrise $\mathcal{T}$.

The definition presupposes that $\mathcal{T}_d$ is a topology in the sense of Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison, and it is. By 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 both $\varnothing$ and $X$ are open in $(X,d)$, which is (T1), and by Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed the family $\mathcal{T}_d$ is closed under arbitrary unions, which is (T2), and under intersections of $n \ge 1$ members, which contains (T3). So every metric space is a topological space, and the metric-space development of this library is a special case of the present one.

The standard local notions in the two developments agree after translating their neighbourhood conventions. Let $d$ be a metric on $X$ and give $X$ the topology $\mathcal{T}_d$.

• Neighbourhoods and balls. 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 uses "neighbourhood" for an open set containing $x$, whereas Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open also allows a non-open superset of such a set. Thus the two collections are not literally equal, but the open metric neighbourhoods are cofinal in the broader neighbourhood filter. The balls $B(x,r)$, $r > 0$ (Open ball, closed ball and sphere in a metric space), are open (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed) and form a neighbourhood base at $x$: any neighbourhood $N$ contains an open $U \ni x$, hence a ball around $x$ by 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. The balls of radius $1/n$ alone already suffice (The balls $B(x, 1/n)$, $n \ge 1$, form a countable neighbourhood base at $x$, so every metric space is first countable).

• Interior, closure, boundary. Interior, closure, boundary, limit point, isolated point and dense subset of a metric space defines them by the same conditions with balls in place of neighbourhoods, and the previous bullet makes the two conditions equivalent; the metric closure is the smallest closed superset (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), which is the definition used here (Interior, closure, boundary, exterior, derived set and isolated point in a topological space). So the two closures, the two interiors and the two boundaries are the same three operations.

• Convergence. $x_k \to p$ in the sense of Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure is "eventually in every neighbourhood of $p$", and by the first bullet this is "eventually in every ball around $p$", which is Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$.

• Continuity. $\varepsilon$-$\delta$ continuity at $a$ (Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form) says that every ball around $f(a)$ has a ball around $a$ mapped into it, which by the first bullet is continuity at $a$ in the sense of Continuity of a map of topological spaces at a point and globally.

• Subspaces. For $A \subseteq X$ the subspace topology $\{\, U \cap A : U \in \mathcal{T}_d \,\}$ of Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace is exactly the metric topology of the subspace metric $d_A$ (Isometry, isometric embedding, and the subspace metric on a subset), so the two uses of the phrase subspace topology in this library name one thing. Indeed $B_A(a,r) = B_X(a,r) \cap A$: a trace $U \cap A$ is $d_A$-open, since each of its points $a$ has $B_X(a,r) \subseteq U$ and hence $B_A(a,r) \subseteq U \cap A$; and a $d_A$-open $W$ is the trace of $U := \bigcup \{\, B_X(a,r) : a \in W,\ r > 0,\ B_X(a,r) \cap A \subseteq W \,\}$, which is $d_X$-open, satisfies $U \cap A = W$, and involves no choice principle, the union being taken over a set of pairs rather than over a selection.

Consequently the metric-space notions of interior, closure, boundary, density, convergence, continuity and subspace agree with the topological notions here, and statements about them transfer once a metric is named. For neighbourhoods the transfer uses the explicit convention change above: a metric-page neighbourhood is an open topological neighbourhood, while every topological neighbourhood contains one.

Metrizability is a topological property; the metric is not part of it. If $h : X \to Y$ is a homeomorphism (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological) and $d$ metrises $X$, then $d'(y_1,y_2) := d(h^{-1}(y_1), h^{-1}(y_2))$ is a metric on $Y$ — the three axioms transfer along the bijection $h^{-1}$ — and $h[B_d(x,r)] = B_{d'}(h(x), r)$, so $h$ carries a basis of $\mathcal{T}_d$ to a basis of $\mathcal{T}_{d'}$ and $\mathcal{T}_{d'} = h[\mathcal{T}_d] = \mathcal{T}_Y$. Hence $Y$ is metrizable. The metric itself, however, is not determined by the topology: two metrics on one set may induce the same topology without agreeing, which is exactly topological equivalence (Topologically, uniformly and Lipschitz equivalent metrics on a set), and properties of a metric that are not properties of its topology, boundedness among them, are therefore not properties of a metrizable space.

Two things every metrizable space has. It is Hausdorff: distinct points have disjoint open neighbourhoods, by Distinct points of a metric space have disjoint balls around them applied to any metric inducing the topology. And it is first countable (First countable space: a countable neighbourhood base at every point), by The balls $B(x, 1/n)$, $n \ge 1$, form a countable neighbourhood base at $x$, so every metric space is first countable. Either failure is therefore an obstruction to metrizability, and this page uses the first of them to exhibit a topology induced by no metric.

Sequential limits in a metrizable space are unique, so the notation $\lim_k x_k$ is available there. In a metric space a sequence has at most one limit (A sequence in a metric space has at most one limit), and by the agreement of convergence above that uniqueness is a statement about the topology alone; so within a metrizable space, and only there, this page writes $\lim_k x_k$ in the ordinary way. In a general space the symbol is unavailable (Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure).

The usual topology of $\mathbb{R}$. The absolute value makes $\mathbb{R}$ a metric space under $d_{\mathbb{R}}(x,y) = |x-y|$, its open balls are the bounded open intervals, and the resulting metric topology is what claim 3 of 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 names the usual topology of $\mathbb{R}$. That is the topology meant by the phrase throughout these two pages, and $\mathbb{R}$ carrying it is metrizable by definition. Every statement about it on these two pages is proved from the metric $d_{\mathbb{R}}$ and the bridge above, and no example re-derives any of it.

• The usual topology of $\mathbb{R}$ is not a second notion alongside the order-native development built earlier in this library. Which results on this page use the order of $\mathbb{R}$ and therefore have no general-topological analogue records that the two collections of open subsets of $\mathbb{R}$ — the one defined from balls and the one defined from order-neighbourhoods — are literally the same collection, and hence that interior, closure, boundary, limit point, density and sequential convergence agree on the two sides. That identification is quoted here for orientation only; the order-topology example on the companion page is where the order-native description is used.

• A metrizable space comes with many metrics and no canonical one. The statement "$X$ is metrizable" asserts existence, and every argument that uses a metric must name one first. Where two metrics are compared, the vocabulary is that of Topologically, uniformly and Lipschitz equivalent metrics on a set: Lipschitz, uniform and topological equivalence, of which only the last is visible to the topology.

• Metrization theorems are not proved here. Necessary conditions are easy — Hausdorff, first countable — and sufficient ones require separation and countability axioms that this page does not develop. Nothing below asserts that a space is metrizable except by exhibiting a metric.

• Not every topology is metrizable, and the cheapest witness is the indiscrete topology on a two-point set, which is not Hausdorff. That is recorded on this page as a false statement and witnessed on the companion page.