Topology — An Interactive Introduction

1 Topological Space

A space stripped down to its bare essence: just a set of points and a rule about which subsets count as "open."

Definition. A topological space is a set X together with a collection 𝜏 of subsets of X (the open sets) satisfying:

∅ and X are both in 𝜏.
Any union (even infinite) of sets in 𝜏 is in 𝜏.
Any finite intersection of sets in 𝜏 is in 𝜏.

The pair (X, 𝜏) is the topological space. Topology is "geometry without distance" — bending and stretching are allowed; tearing and gluing are not.

On the right, the same 4-point set X = {a, b, c, d} carries different topologies. Cycle through them and watch which subsets are declared open. A valid topology must always contain ∅ and X, and be closed under unions & finite intersections.

2 Definition of Open Sets

In the familiar plane, "open" means every point has a little breathing room — wiggle room in all directions without leaving the set.

Metric definition. A set U ⊆ ℝⁿ is open if for every point p ∈ U there exists a radius ε > 0 such that the entire open ball B(p, ε) lies inside U. Intuitively: no point of an open set sits on its own edge.

Move your cursor (or drag) inside the disk. A point near the middle has room for a full ε-ball — it's an interior point. Drag toward the boundary and the largest fitting ball shrinks to nothing: that point is not interior, which is exactly why the closed disk's boundary keeps it from being open.

ε = —

Green ball fully inside → interior point. Red → the ball pokes out past the edge.

3 Examples of Open Sets

Open, closed, or neither? Click each shape to test it against the "breathing room" rule.

① open interval ② open disk ③ closed disk ④ half-open Click a shape →

Each example shows whether boundary points are included (solid edge = closed boundary) or excluded (dashed edge = open boundary).

Open. (a, b) in ℝ; the open ball B(p,ε); any union of open balls; the whole space ℝⁿ; the empty set ∅.

Not open. A closed interval [a, b]; a single point {p}; a half-open [a, b); any set containing a boundary point of itself.

4 Continuous Mapping

Forget epsilons and deltas — topology defines continuity purely through open sets.

Definition. A map f : X → Y is continuous if the preimage f⁻¹(V) of every open set V ⊆ Y is open in X. Equivalently in ℝ: nearby inputs map to nearby outputs — the graph has no jumps.

The animation feeds a sliding open interval (top axis, domain) through a function and shows its image on the output axis. For a continuous map the image stays a single connected open blob. Toggle a jump discontinuity and watch an open input produce a "broken" preimage — the hallmark of a non-continuous map.

5 Homeomorphism

When two spaces are "the same" to a topologist: continuously deformable into one another, no cutting, no gluing.

Definition. A homeomorphism is a bijection f : X → Y that is continuous and whose inverse is also continuous. If one exists, X and Y are homeomorphic — topologically identical. The classic punchline: a coffee mug is a doughnut, because each has exactly one hole.

Watch a coffee mug morph continuously into a torus (doughnut). The single hole — the handle becomes the doughnut's hole — is preserved throughout. No point is ever torn from its neighbors, so the transformation is a genuine homeomorphism.

mug

Genus (number of holes) = 1 the whole way → topologically invariant.

6 Compactness

A subtle finiteness: no matter how you cover the space with open sets, a finite handful already suffices.

Definition. A space is compact if every open cover has a finite subcover. In ℝⁿ (Heine–Borel) this is exactly closed and bounded. Compactness is what lets continuous functions attain their max and min.

The strip is covered by infinitely many overlapping open intervals. On a compact (closed, bounded) interval you can throw most of them away and keep only finitely many that still cover everything — press "Extract finite subcover." On the open-ended ray, the cover near the missing endpoint needs ever-smaller sets: no finite subcover exists.

Infinite open cover shown.

7 Connectivity

A space is connected if it's "all one piece" — it can't be split into two separate open chunks.

Definition. A space X is connected if it cannot be written as the union of two disjoint, non-empty open sets. Equivalently, you can travel between any two points without ever leaving the space (path-connected, in nice cases).

Click "Cut" to slice the blob into two disjoint open pieces — now it's disconnected. A travelling probe tries to walk from the left point to the right point along the space; when a gap appears it gets stuck, proving the two halves are genuinely separated.