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Given a metric d on X, the metric topology consists of all unions of open balls B(x,ε) = {y : d(x,y) < ε}. Open balls form a basis for this topology. Not every topology comes from a metric (metrization theorems characterize which do). Metrics provide explicit, computable topologies on ℝⁿ and function spaces.
A metric gives you a way to measure distance: d(x, y) satisfies non-negativity, symmetry, and the triangle inequality. From your prerequisites you know what a metric space is and what open sets look like. The goal here is to see precisely how a metric *generates* a topology — how the distance function produces a family of open sets that satisfies all the topological axioms.
The construction begins with open balls: B(x, ε) = {y ∈ X : d(x, y) < ε}. Think of this as all points strictly within distance ε of x — the interior of a ball of radius ε centered at x. On the real line, B(x, ε) = (x−ε, x+ε), an open interval. In ℝ², it is an open disk. In a discrete metric space (where d(x,y) = 1 for all x ≠ y), B(x, 1/2) = {x}, a single point. The metric topology is then defined as the collection of all sets that can be written as unions of open balls. A set U is open in the metric topology if for every point x ∈ U, some open ball B(x, ε) is contained in U — equivalently, every point of U has a little breathing room inside U. Verifying that this collection satisfies the topological axioms (closed under arbitrary unions, finite intersections, contains ∅ and X) is a standard exercise using the triangle inequality.
The open balls form a basis for this topology, meaning every open set is a union of open balls — but the open balls themselves need not form a topology, because finite intersections of open balls are not generally open balls. The concept of a basis is important: you rarely specify a topology by listing all open sets (there can be uncountably many). Instead, you specify a basis and declare a set open if it's a union of basis elements. The metric provides such a basis for free.
A central point is that different metrics can generate the same topology. On ℝⁿ, the Euclidean metric, the taxicab metric (sum of coordinate differences), and the maximum metric all generate the same topology — the standard one — even though the shapes of their open "balls" differ. Two metrics that generate the same topology are called topologically equivalent. This shows that topology abstracts away the specific geometry (exact distances) and retains only the qualitative structure (which sets are open). The converse question — which topologies arise from some metric — is answered by metrization theorems: not every topology is metrizable, and conditions like second-countability and regularity characterize which ones are. The metric topology sits at a middle level of generality: richer than abstract topology (because you have explicit distances), but not as rigid as Euclidean geometry (because distances can be exotic). Most spaces you encounter in analysis — function spaces like C([0,1]) with the sup-norm, sequence spaces like ℓ², manifolds — are metric spaces, making this the dominant setting for applied topology.