Topological Spaces

In this lesson you’ll meet the definition of a topological space, learn why unions are unrestricted while intersections are only finite, work through the standard topologies, and see how they compare in fineness.

A topology on a set $X$ is a collection $\tau$ of subsets of $X$ such that

1. $\varnothing \in \tau$ and $X \in \tau$
2. $\tau$ is closed under arbitrary unions
3. $\tau$ is closed under finite intersections

The pair $(X,\tau)$ is a topological space, and the members of $\tau$ are the open sets.

That is the whole definition. No distance, no coordinates, no algebra. You simply declare which subsets are open, and the three rules are the only constraints on that declaration.

The figure tests three candidate families on a three-element set by brute force over all unions and intersections. Two pass. The third contains $\{a\}$ and $\{b\}$ but not their union $\{ab\}$, so it fails axiom 2 - and the figure names the offending union rather than just reporting failure.

The most-asked question about the definition: why arbitrary unions but only finite intersections?

Because with metric spaces as the model, that is exactly what is true. In $\mathbb{R}$ consider

$$\bigcap_{n=1}^{\infty}\left(-\tfrac1n,\ \tfrac1n\right) = \{0\}$$

Every set in the intersection is open, and the result is a single point, which is not open. So requiring arbitrary intersections would exclude $\mathbb{R}$ itself from being a topological space.

Arbitrary unions cause no such trouble: a union of open balls always has a ball around each of its points, since each point came from one of the balls.

The asymmetry is not a stylistic choice; it is forced by the intended examples.

A set is closed if its complement is open.

By De Morgan, the axioms dualize: closed sets are closed under arbitrary intersections and finite unions. Note the reversal.

Three warnings that cause most early confusion:

• Open and closed are not opposites. A set can be both, neither, or one.
• $\varnothing$ and $X$ are always both open and closed.
• “Not open” does not mean closed. In $\mathbb{R}$, the interval $[0,1)$ is neither.

A set that is both open and closed is called clopen, and how many clopen sets a space has turns out to measure its connectedness - that is lesson 8.

Worth knowing cold, because they are the test cases for everything.

The discrete topology. $\tau =$ all subsets. Every set is open, and also closed. It comes from the discrete metric. This is the finest possible topology.

The trivial (indiscrete) topology. $\tau = \{\varnothing, X\}$ and nothing else. The coarsest possible topology.

The standard topology on $\mathbb{R}$. Open sets are the unions of open intervals. This is the one implied whenever no topology is named.

The cofinite topology. Open sets are $\varnothing$ together with every set whose complement is finite. It is a topology on any set, and on an infinite set it is genuinely strange: any two nonempty open sets intersect, so no two points can be separated.

The Zariski topology. In algebraic geometry, closed sets are the zero sets of polynomials. On a line over a field this is essentially the cofinite topology, and it is central to the subject even though it is far from being metric.

The lower limit topology on $\mathbb{R}$. Generated by half-open intervals $[a,b)$. Strictly finer than the standard topology, and a reliable source of counterexamples.

Any metric space. As in the previous lesson, open sets are those containing a ball around each of their points. A topology arising this way is called metrizable, and not all topologies are.

If $\tau_1\subseteq\tau_2$ on the same set, then $\tau_2$ is finer and $\tau_1$ is coarser.

Finer means more open sets, which means more distinctions between points, which means it is easier for a set to be open and harder for a function into the space to be continuous.

Every topology on $X$ sits between the trivial and the discrete:

$$\{\varnothing,X\} \ \subseteq\ \tau \ \subseteq\ \mathcal{P}(X)$$

Not all pairs are comparable, though. Two topologies can each contain sets the other lacks.

The number of distinct topologies grows explosively: 1 on a one-point set, 4 on a two-point set, 29 on a three-point set, 355 on four points, 6942 on five. There is no known formula.

A neighbourhood of $x$ is any set containing an open set that contains $x$.

Some books insist a neighbourhood be open itself; the difference rarely matters. The useful reformulation:

$U$ is open if and only if $U$ is a neighbourhood of each of its points.

That is the statement that most closely echoes the metric definition, and it is how the two pictures line up.

Example 1: Is $\{\varnothing, \{a\}, \{b\}, \{a,b,c\}\}$ a topology on $\{a,b,c\}$?

Solution. Check the axioms.

Axiom 1: contains $\varnothing$ and $X$ ✓

Axiom 3, intersections: $\{a\}\cap\{b\}=\varnothing$ ✓, and all others are fine ✓

Axiom 2, unions: $\{a\}\cup\{b\} = \{a,b\}$, which is not in the collection ✗

Answer: not a topology. ∎

Example 2: Is $\{\varnothing, \{a\}, \{a,b\}, \{a,b,c\}\}$ a topology?

Solution. This family is a chain, each set contained in the next. Any union of members is the largest of them, and any intersection is the smallest, so both are in the family.

Answer: a topology. ∎

Nested chains are automatically topologies, which is a handy shortcut when checking examples.

Example 3: Verify the cofinite topology on $\mathbb{Z}$ is a topology.

Solution. $\varnothing$ is open by decree and $\mathbb{Z}$ is open since its complement is empty, hence finite ✓

Unions: a union of cofinite sets has complement equal to the intersection of finite sets, which is finite ✓

Intersections: a finite intersection of cofinite sets has complement the union of finitely many finite sets, again finite ✓

$$\text{a topology}$$

Note the finiteness in axiom 3 is genuinely needed: an infinite intersection of cofinite sets, such as all sets missing one particular integer, can have infinite complement. ∎

Example 4: Show $\{0\}$ is not open in $\mathbb{R}$ despite being an intersection of open sets.

Solution. We have $\bigcap_{n\ge1}(-\frac1n,\frac1n) = \{0\}$, since any nonzero $x$ is excluded once $\frac1n<|x|$.

Every set in the intersection is open, but $\{0\}$ contains no interval, so it is not open.

This is precisely why axiom 3 says finite. ∎

Example 5: In the cofinite topology on an infinite set, can two nonempty open sets be disjoint?

Solution. No. If $U$ and $V$ are nonempty and open, their complements are finite, so

$$X\setminus(U\cap V) = (X\setminus U)\cup(X\setminus V)$$

is finite. Since $X$ is infinite, $U\cap V$ must be nonempty.

Every two nonempty open sets meet. ∎

This is the failure of the Hausdorff condition, and it is why the cofinite topology on an infinite set is not metrizable - a fact we prove in lesson 13.

Example 6: Which sets are clopen in $\mathbb{R}$ with the standard topology?

Solution. Only $\varnothing$ and $\mathbb{R}$.

Suppose $A$ is nonempty, clopen, and not all of $\mathbb{R}$. Then $A$ and its complement are both open, nonempty, and partition $\mathbb{R}$ - which contradicts connectedness of $\mathbb{R}$, proved in lesson 8.

$$\text{clopen sets: } \varnothing \text{ and } \mathbb{R}$$

∎

Contrast the discrete topology, where every set is clopen. The count of clopen sets is a direct measure of how disconnected a space is.

Example 7: Compare the standard and lower limit topologies on $\mathbb{R}$.

Solution. Every open interval $(a,b)$ is a union of half-open intervals:

$$(a,b) = \bigcup_{n}\left[a+\tfrac1n,\ b\right)$$

so every standard-open set is lower-limit-open, giving $\tau_{\text{std}}\subseteq\tau_{\text{ll}}$.

The containment is strict, because $[0,1)$ is lower-limit-open and not standard-open.

$$\tau_{\text{ll}} \text{ is strictly finer}$$

∎

Type systems and program semantics. Domain theory, used to give meaning to recursive programs, is built on the Scott topology, where open sets encode “observable in finite time.” Non-Hausdorff topologies are the norm there and the metric picture is useless.

Databases and logic. The Stone topology on the set of models of a theory turns logical compactness into topological compactness, which is where the name came from.

Algebraic geometry. The Zariski topology is the working topology of the field, and it is far from metric: it is not Hausdorff, and its open sets are enormous.

Digital images. Digital topology treats a pixel grid as a topological space in order to define connectedness of regions consistently, which matters for segmentation and boundary tracing.

Concurrency. Directed topological spaces model the possible execution paths of concurrent programs, and deadlock detection becomes a question about holes.

Sensor networks. Coverage without coordinates is a topological question, answered from connectivity data alone, which is exactly the setting where distance is unavailable.