02.01.02 · analysis / topology

Continuous map

shipped3 tiersLean: none

Anchor (Master): Munkres §18; Willard §6

Intuition [Beginner]

A continuous map sends nearby points to nearby points. It does not tear a connected piece apart or make a sudden jump.

For a rubber-sheet picture, imagine drawing a curve and then bending or stretching the sheet without cutting it. Points that were close remain controlled by the motion. That is the geometric intuition behind continuity.

Topology defines continuity without measuring distance. Instead of asking for small distance changes, it asks that open regions pull back to open regions.

Visual [Beginner]

A continuous map carries a flexible shape into another space without jumps. Open target windows pull back to open source windows.

The backward direction is important: continuity is tested by preimages of open sets.

Worked example [Beginner]

The map from a line to a plane given by sending a number $t$ to the point on a circle at angle $t$ is continuous. As $t$ changes a little, the point on the circle moves a little.

A step function is different. It stays at one height for negative inputs and jumps to another height for positive inputs. Near the jump, tiny changes in input can create a sudden change in output.

What this tells us: continuity is the rule that forbids sudden jumps.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ and $Y$ be topological spaces. A function $f : X \to Y$ is continuous if for every open set $V \subseteq Y$ , the preimage

f^{- 1} (V) = {x \in X : f (x) \in V}

is open in $X$ ^{[Munkres §18]}.

A homeomorphism is a bijective continuous map whose inverse is continuous. Homeomorphic spaces are the same from the viewpoint of topology.

For metric spaces, this definition agrees with the epsilon-delta definition: $f : (X, d_{X}) \to (Y, d_{Y})$ is continuous at $x$ if every $ε > 0$ admits a $δ > 0$ such that $d_{X} (x, x^{'}) < δ$ implies $d_{Y} (f (x), f (x^{'})) < ε$ .

Key theorem with proof [Intermediate+]

Theorem (Composition of continuous maps). If $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces, then $g \circ f : X \to Z$ is continuous.

Proof. Let $W \subseteq Z$ be open. Since $g$ is continuous, $g^{- 1} (W)$ is open in $Y$ . Since $f$ is continuous, $f^{- 1} (g^{- 1} (W))$ is open in $X$ .

But

(g \circ f)^{- 1} (W) = f^{- 1} (g^{- 1} (W)) .

Therefore the preimage of every open set under $g \circ f$ is open, so $g \circ f$ is continuous. $□$

Bridge. The construction here builds toward 03.08.04 (classifying space), where the same data is upgraded, and the symmetry side is taken up in 03.05.01 (principal bundle). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none is recorded because this unit needs the curriculum's notation bridge between topological continuity, metric continuity, homeomorphism, pullback constructions, and classifying maps before Mathlib statements are exposed.

Advanced results [Master]

Continuity can be expressed by closure, neighborhoods, nets, or filters. For general topological spaces, filter and net characterizations replace the sequence-based tests that are sufficient in first-countable spaces ^{[Willard §6]}.

The preimage definition is functorial: topological spaces and continuous maps form a category. Products, pullbacks, quotient maps, and bundle constructions use this categorical behavior.

Synthesis. Continuous maps are the morphisms of the category of topological spaces: they preserve the open-set structure and therefore preserve limits, compactness, connectedness, and separation properties. The epsilon-delta characterisation for metric spaces generalises to the inverse-image definition for arbitrary topological spaces, revealing continuity as a property of how structure is preserved rather than a property of distances. This categorical perspective connects topology to algebra (continuous maps induce homomorphisms of fundamental groups and cohomology rings), to analysis (continuous linear maps between normed spaces are the bounded operators, continuous functions on compact spaces form C*-algebras), and to geometry (homeomorphisms classify surfaces and manifolds up to topological type, while diffeomorphisms carry the additional smooth structure).

Full proof set [Master]

The identity map is continuous because preimages of open sets are themselves. The composition theorem proves closure under composition. Together these two facts prove that topological spaces with continuous maps form a category.

For homeomorphisms, continuity in both directions makes the bijection an isomorphism in this category. The proof in Exercise 7 shows that a homeomorphism is exactly a bijection preserving open sets in both directions.

Connections [Master]

Continuous maps depend on topological spaces 02.01.01 and feed metric spaces 02.01.05, classifying spaces 03.08.04, and bundle pullback constructions 03.05.01. Smooth maps between manifolds are continuous maps with differentiability added 03.02.01.
K-theory 03.08.01 is contravariant for continuous maps through pullback of vector bundles.

Historical & philosophical context [Master]

The open-set definition of continuity became standard with the development of point-set topology. Munkres presents it as the central definition because it works beyond metric spaces ^{[Munkres §18]}.

Willard develops equivalent formulations through neighborhoods and filters, which are useful in general topology and functional analysis ^{[Willard §6]}.

Bibliography [Master]

James Munkres, Topology, §18. ^{[Munkres §18]}
Stephen Willard, General Topology, §6. ^{[Willard §6]}