Topology Space

Every notion of “closeness” you have met so far — convergence, limits, continuity — was defined in terms of a distance function. That works beautifully in $\mathbb{R}^n$, but many spaces that arise naturally in mathematics and computer science have no sensible notion of distance at all. Topological spaces let you keep all of that theory while throwing out the distance function entirely.

The key insight from metric spaces

In a metric space $(X, d)$, a set $U \subseteq X$ is called open when every point in $U$ has an open ball around it that fits entirely inside $U$:

$$U \text{ is open} \iff \forall x \in U,\ \exists r > 0 : B(x, r) \subseteq U.$$

Once you have this collection of open sets, you can re-express every major concept — convergence, continuity, connectedness — using only membership in open sets. The metric itself disappears from the statements.

This is the insight: the open sets are doing all the work. The distance function is just one way to generate them.

The axioms of a topology

Pause and ask: what properties does the collection of open sets in a metric space actually have? You can verify three facts for any metric space:

1. The empty set $\varnothing$ and the whole space $X$ are both open.
2. Any union of open sets (even an infinite one) is open.
3. Any finite intersection of open sets is open.

Note that infinite intersections are not required to be open: in $\mathbb{R}$, the intersection $\bigcap_{n=1}^{\infty} \left(-\tfrac{1}{n}, \tfrac{1}{n}\right) = \{0\}$ is a single point, which is closed, not open.

These three facts are now taken as axioms. A topology throws away everything except what these three properties need.

Definition. A topology on a set $X$ is a collection $\tau \subseteq \mathcal{P}(X)$ of subsets of $X$ satisfying:

$$\text{(T1)} \quad \varnothing \in \tau \text{ and } X \in \tau, \tag{1}$$ $$\text{(T2)} \quad \text{if } U_\alpha \in \tau \text{ for all } \alpha \in I, \text{ then } \bigcup_{\alpha \in I} U_\alpha \in \tau, \tag{2}$$ $$\text{(T3)} \quad \text{if } U_1, \ldots, U_n \in \tau, \text{ then } U_1 \cap \cdots \cap U_n \in \tau. \tag{3}$$

The pair $(X, \tau)$ is called a topological space. The elements of $\tau$ are called the open sets of the topology.

Every metric space is automatically a topological space: take $\tau$ to be the collection of all sets that are open in the metric sense — the three axioms hold, as you verified above.

Three fundamental examples

The discrete topology

Let $X$ be any set, and let $\tau \coloneqq \mathcal{P}(X)$ — every subset of $X$ is declared open. This is the discrete topology. You can check T1–T3 immediately: unions and finite intersections of subsets are subsets.

The discrete topology is the “finest” possible topology on $X$: it has the most open sets. It corresponds to the discrete metric ($d(x, y) = 1$ for $x \neq y$), where every singleton $\{x\}$ is an open ball.

The indiscrete topology

At the other extreme, let $\tau \coloneqq \{\varnothing, X\}$ — only the empty set and the whole space are open. This is the indiscrete topology (also called the trivial topology). T1 is satisfied by definition; T2 and T3 reduce to easy cases with only two sets.

The indiscrete topology is the “coarsest” possible topology: it has the fewest open sets. It is not generated by any metric on more than one point.

The metric topology

Given a metric space $(X, d)$, the collection

$$\tau_d \coloneqq \{ U \subseteq X : \forall x \in U,\ \exists r > 0 : B(x, r) \subseteq U \}$$

is a topology on $X$, called the metric topology induced by $d$. This is the bridge between metric spaces and topology spaces: every metric space “is” a topological space via this construction, but not every topological space comes from a metric.

Open and closed sets

A subset $C \subseteq X$ is closed if its complement $X \setminus C$ is open.

Note carefully: “closed” is not the opposite of “open” in common parlance. Sets can be:

• Open but not closed: $(0, 1)$ in $\mathbb{R}$ with the standard metric topology.
• Closed but not open: $[0, 1]$ in the same topology.
• Both open and closed (clopen): $\varnothing$ and $X$ are always clopen. In a disconnected space, other clopen sets exist too.
• Neither open nor closed: $[0, 1)$ in $\mathbb{R}$.

The axioms T1–T3, translated to closed sets via De Morgan’s laws, say:

• $\varnothing$ and $X$ are closed.
• Any finite union of closed sets is closed.
• Any intersection of closed sets (even infinite) is closed.

This is a useful dual picture that you will see repeatedly.

Comparing topologies on the same set

Two different topologies can live on the same set $X$. If $\tau_1 \subseteq \tau_2$ — every open set of $\tau_1$ is also open in $\tau_2$ — then $\tau_2$ is said to be finer than $\tau_1$ (and $\tau_1$ is coarser than $\tau_2$). Finer means more open sets; more open sets means a more refined notion of “closeness.”

The discrete topology is finer than every other topology on $X$. The indiscrete topology is coarser than every other topology on $X$.

In general, two topologies on the same set need not be comparable: neither may be a subset of the other.

Why abandon the metric?

The payoff is generality. Here are two spaces that have natural topologies but no obvious metric:

• The Zariski topology on an algebraic variety, where open sets are complements of the zero-sets of polynomials. This is central to algebraic geometry and, via algebraic $K$-theory, eventually touches type systems and programming language semantics.
• The Scott topology on a partially ordered set, where open sets capture “upward-closed sets closed under directed joins.” This topology is the mathematical foundation for denotational semantics — the very framework used to give a precise meaning to recursive programs.

By working in the generality of topological spaces you can talk about continuity, convergence, and compactness in all of these settings with a single set of definitions.

Summary

• A topological space $(X, \tau)$ is a set $X$ together with a collection $\tau$ of open sets satisfying three axioms: $\varnothing$ and $X$ are open (T1); arbitrary unions of open sets are open (T2); finite intersections of open sets are open (T3).
• Every metric space gives rise to a topological space by taking all metrically-open sets as $\tau$.
• The discrete topology ($\tau = \mathcal{P}(X)$) is the finest; the indiscrete topology ($\tau = \{\varnothing, X\}$) is the coarsest.
• A set is closed when its complement is open; sets can be open, closed, both (clopen), or neither.
• One topology is finer than another when it contains more open sets.
• The abstraction strips distance away, leaving only the open-set structure — which is enough to define continuity, convergence, and connectedness in far more general settings.