Analysis begins with convergence: a sequence $x_1, x_2, \ldots$ in $\mathbb{R}$ converges to $L$ if $|x_n - L| \to 0$. In $\mathbb{R}^n$, the absolute value becomes the Euclidean distance, and the same definition carries over. But many of the most important spaces in mathematics — spaces of [functions](/page/Function), [distributions](/page/Distribution), measures — are infinite-dimensional and carry no single canonical distance. Worse, some natural notions of convergence ([weak convergence](/page/Weak%20Convergence), convergence in distribution) cannot be described by any metric at all. The resolution is to abandon distance as the primitive concept and axiomatise the structure that distance provides: the collection of "[open sets](/page/Open%20Set)." This is the subject of **topology**.

This page develops the foundations of point-set topology: the axioms for a topological space, bases, [closed sets](/page/Closed%20Set) and closure, [continuity](/page/Continuity) and homeomorphisms, separation, compactness, and connectedness. The more specialised structures that analysis requires — [normed spaces](/page/Normed%20Vector%20Space), [topological vector spaces](/page/Topological%20Vector%20Space), metrizable spaces — are developed on their own pages.

[motivation] ## Motivation ### [Metric Spaces](/page/Metric%20Space): What Distance Gives You In a metric space $(X, d)$, the distance function $d: X \times X \to [0, \infty)$ provides a complete toolkit for analysis: open balls $B(x, r) = \{y \in X : d(x, y) < r\}$, convergence ($x_n \to x$ iff $d(x_n, x) \to 0$), continuity ($f$ is continuous at $x$ iff $d(f(x_n), f(x)) \to 0$ whenever $d(x_n, x) \to 0$), and compactness. Every metric space carries a natural topology — the collection of sets that are unions of open balls — and the metric concepts of convergence and continuity depend only on this topology, not on the specific values of $d$. This is the starting point: if the metric is dispensable and only the open sets matter, why insist on having a metric at all? ### What Metrics Cannot Do The need to go beyond metrics arises in concrete situations. Consider the space $C([0,1])$ of continuous functions on the unit interval. The supremum metric $d_\infty(f, g) = \sup_{x \in [0,1]} |f(x) - g(x)|$ captures [uniform convergence](/page/Uniform%20Convergence) — but what about pointwise convergence, where $f_n(x) \to f(x)$ for each $x$? This is a natural notion of convergence, but it is not given by any metric on $C([0,1])$: the topology of pointwise convergence on an uncountable index set is not metrizable. Similarly, the [weak* topology](/page/Weak*%20Topology) on the dual of an infinite-dimensional [Banach space](/page/Banach%20Space) is typically not metrizable on the full space, only on bounded subsets. Even when a metric exists, the same set can carry multiple metrics that produce identical open sets. On $\mathbb{R}$, the metrics $d_1(x,y) = |x-y|$ and $d_2(x,y) = \min(|x-y|, 1)$ give the same open sets, the same convergent [sequences](/page/Sequence), and the same continuous functions. The metric carries redundant information; the topology is the intrinsic structure. ### The Correct Abstraction What properties must a collection of "open sets" satisfy? Examining what holds in every metric space, one extracts three axioms: (i) the empty set and the whole space are open, (ii) arbitrary unions of open sets are open, and (iii) finite intersections of open sets are open. These are the axioms of a topology. The insistence on *arbitrary* unions but only *finite* intersections is not arbitrary — in a metric space, the intersection $\bigcap_{n=1}^\infty (-1/n, 1/n) = \{0\}$ is a single point, which is not open in $\mathbb{R}$, while any union of open balls is always open. [/motivation]

## Definition

The axioms of a topology distil from metric space theory exactly the structure needed for convergence, continuity, and all related concepts. The definition asks only for a collection of subsets satisfying closure under arbitrary unions and finite intersections.

[definition:Topology] Let $X$ be a [set](/page/Set). A **topology** on $X$ is a collection $\tau \subseteq \mathcal{P}(X)$ of subsets of $X$ satisfying: 1. $\varnothing \in \tau$ and $X \in \tau$. 2. If $\{U_i\}_{i \in I}$ is any collection of elements of $\tau$, then $\bigcup_{i \in I} U_i \in \tau$. 3. If $U_1, \ldots, U_n \in \tau$, then $U_1 \cap \cdots \cap U_n \in \tau$. The pair $(X, \tau)$ is called a **topological space**. The elements of $\tau$ are called **open sets**. [/definition]

The axioms encode two asymmetries. Unions are unrestricted but intersections must be finite — infinite intersections of open sets need not be open, as $\bigcap_{n=1}^\infty (-1/n, 1/n) = \{0\}$ demonstrates. The axioms say nothing about complements; the behaviour of closed sets must be deduced, and it turns out to be dual: arbitrary intersections of closed sets are closed, but only finite unions.

[example:Standard Topology On R] The **standard (Euclidean) topology** on $\mathbb{R}$ consists of all sets $U \subseteq \mathbb{R}$ such that for every $x \in U$, there exists $\varepsilon > 0$ with $(x - \varepsilon, x + \varepsilon) \subseteq U$. This is the topology induced by the metric $d(x,y) = |x-y|$: a set is open if and only if it is a union of open intervals. Verification of the axioms: the empty union gives $\varnothing$; $\mathbb{R}$ is itself open; arbitrary unions of open sets remain open; and a finite intersection of open sets is open because the minimum of finitely many positive $\varepsilon$-values is still positive. [/example]

[example:Extreme Topologies] Every set $X$ carries two extreme topologies. The **discrete topology** $\tau = \mathcal{P}(X)$ declares every subset open: every function out of $X$ is continuous, and convergence means eventual constancy. The **indiscrete (trivial) topology** $\tau = \{\varnothing, X\}$ declares only the empty set and $X$ open: every sequence converges to every point. The discrete topology has "too many" open sets for interesting analysis; the indiscrete topology has too few. All useful topologies lie between these extremes. [/example]

## Bases and the Generation of Topologies

In practice, topologies are rarely specified by listing all open sets. The standard topology on $\mathbb{R}$ is described by saying "a set is open if every point has an open-interval neighbourhood inside it" — the open intervals *generate* the topology. This idea generalises: a smaller collection of sets, called a **basis**, can determine the full topology by taking arbitrary unions.

### Bases for a Topology

The question is: what conditions must a collection $\mathcal{B}$ satisfy so that "arbitrary unions of elements of $\mathcal{B}$" forms a topology? Two conditions suffice: $\mathcal{B}$ must cover $X$, and any point in the intersection of two basis elements must be contained in a third basis element inside that intersection.

[definition:Basis For A Topology] Let $X$ be a set. A collection $\mathcal{B} \subseteq \mathcal{P}(X)$ is a **basis for a topology** on $X$ if: 1. For every $x \in X$, there exists $B \in \mathcal{B}$ with $x \in B$. 2. If $B_1, B_2 \in \mathcal{B}$ and $x \in B_1 \cap B_2$, there exists $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$. The topology $\tau_\mathcal{B}$ **generated by** $\mathcal{B}$ consists of all sets that are unions of elements of $\mathcal{B}$ (including the empty union $\varnothing$). A set $U$ is open in $\tau_\mathcal{B}$ if and only if for every $x \in U$, there exists $B \in \mathcal{B}$ with $x \in B \subseteq U$. [/definition]

Condition (2) is precisely what is needed to ensure that finite intersections of unions of basis elements are again unions of basis elements — it is the mechanism by which the basis axioms guarantee axiom (3) of a topology.

[example:Basis For The Standard Topology On R] The open intervals $\{(a, b) : a < b\}$ form a basis for the standard topology on $\mathbb{R}$. A strictly smaller basis also works: the open intervals with rational endpoints $\{(p, q) : p < q,\; p, q \in \mathbb{Q}\}$ generate the same topology, since every open interval contains a rational-endpoint sub-interval around each of its points. This countable basis witnesses the **second countability** of $\mathbb{R}$ — a property with profound consequences for [separability](/page/Separable) and metrizability. [/example]

### The Metric Topology

The basis concept explains why every metric space is automatically a topological space. In a metric space $(X, d)$, the open balls $\{B(x, r) : x \in X,\, r > 0\}$ form a basis: any point in $B(x_1, r_1) \cap B(x_2, r_2)$ is contained in a smaller ball inside the intersection (take $r_3 = \min(r_1 - d(x, x_1), r_2 - d(x, x_2))$, which is positive by the triangle inequality). The topology generated by this basis is the **metric topology**, and it is the unique topology in which a sequence converges if and only if $d(x_n, x) \to 0$.

Different metrics on the same set can generate the same topology. Two metrics $d_1$ and $d_2$ on $X$ are called **topologically equivalent** if they generate the same open sets. For instance, on $\mathbb{R}^n$, the metrics induced by the $\ell^1$, $\ell^2$, and $\ell^\infty$ norms are all topologically equivalent — they have different unit balls but the same open sets. A topological space whose topology arises from some metric is called **metrizable**; not every topological space is metrizable, and characterising which ones are is a deep question (answered by the Urysohn metrization theorem and the Nagata-Smirnov theorem).

## Closed Sets, Closure, and Interior