Open and Closed Maps

While continuous maps are defined by the behavior of preimages, open and closed maps are defined by the behavior of images. These notions complement continuity and play essential roles in product topologies, quotient topologies, and the study of homeomorphisms.

Definitions

Definition2.7Open Map

A function $f: X \to Y$ between topological spaces is an open map if for every open set $U \subseteq X$ , the image $f(U)$ is open in $Y$ .

Definition2.8Closed Map

A function $f: X \to Y$ is a closed map if for every closed set $C \subseteq X$ , the image $f(C)$ is closed in $Y$ .

RemarkIndependence from Continuity

Being an open map, a closed map, or a continuous map are three independent conditions. A map can satisfy any combination of these properties.

The projection $\pi_1: \mathbb{R}^2 \to \mathbb{R}$ is continuous and open, but not closed (the image of the closed hyperbola $\{xy = 1\}$ is $\mathbb{R} \setminus \{0\}$ , which is not closed).
A continuous bijection that is also open (or closed) is automatically a homeomorphism.

Examples

ExampleOpen and Closed Maps

Projections are open: The projection $\pi_1: X \times Y \to X$ from a product space is always an open map. If $U \times V$ is a basic open set in $X \times Y$ , then $\pi_1(U \times V) = U$ is open. Since $\pi_1$ preserves unions, $\pi_1$ maps any open set to an open set.
Projections need not be closed: Let $C = \{(x, y) \in \mathbb{R}^2 : xy = 1\}$ . Then $C$ is closed in $\mathbb{R}^2$ , but $\pi_1(C) = \mathbb{R} \setminus \{0\}$ is not closed in $\mathbb{R}$ .
Closed maps from compact spaces: If $X$ is compact and $Y$ is Hausdorff, then every continuous map $f: X \to Y$ is a closed map (see Chapter 5).
Inclusion of a closed subspace: If $A$ is a closed subspace of $X$ , the inclusion $\iota: A \hookrightarrow X$ is a closed map.

Relationship to Homeomorphisms

Theorem2.3Characterization of Homeomorphisms

Let $f: X \to Y$ be a bijection. The following are equivalent:

$f$ is a homeomorphism.
$f$ is continuous and open.
$f$ is continuous and closed.

Proof

(1 $\Rightarrow$ 2): If $f$ is a homeomorphism, then $f$ is continuous. For openness: if $U$ is open in $X$ , then $(f^{-1})^{-1}(U) = f(U)$ , which is open since $f^{-1}$ is continuous.

(2 $\Rightarrow$ 1): We need $f^{-1}$ to be continuous. For any open $U \subseteq X$ , $(f^{-1})^{-1}(U) = f(U)$ , which is open since $f$ is an open map. So $f^{-1}$ is continuous.

(1 $\Leftrightarrow$ 3): Similar argument using closed sets.

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Quotient Maps and Saturation

Definition2.9Saturated Set

Let $f: X \to Y$ be a surjective map. A subset $U \subseteq X$ is saturated with respect to $f$ if $U = f^{-1}(f(U))$ , i.e., $U$ is a union of fibers of $f$ .

Theorem2.4Characterization via Saturated Sets

A surjective continuous map $f: X \to Y$ is a quotient map if and only if $f$ maps saturated open sets to open sets (equivalently, saturated closed sets to closed sets).

ExampleSaturated Sets

Consider the projection $\pi: \mathbb{R}^2 \to \mathbb{R}$ , $\pi(x, y) = x$ . A set $U \subseteq \mathbb{R}^2$ is saturated with respect to $\pi$ if and only if it is a union of vertical lines, i.e., $U = A \times \mathbb{R}$ for some $A \subseteq \mathbb{R}$ .

The open strip $(-1, 1) \times \mathbb{R}$ is saturated and open, with image $(-1, 1)$ open in $\mathbb{R}$ .

Proper Maps

Definition2.10Proper Map

A continuous map $f: X \to Y$ is proper if for every compact set $K \subseteq Y$ , the preimage $f^{-1}(K)$ is compact in $X$ .

Theorem2.5Proper Maps are Closed

If $f: X \to Y$ is a proper map and $Y$ is a locally compact Hausdorff space, then $f$ is a closed map.

ExampleProper Maps

The map $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = x^2$ is proper: if $K \subseteq [0, \infty)$ is compact, say $K \subseteq [0, M]$ , then $f^{-1}(K) \subseteq [-\sqrt{M}, \sqrt{M}]$ is closed and bounded, hence compact.
The projection $\pi_1: \mathbb{R}^2 \to \mathbb{R}$ is not proper: $\pi_1^{-1}([0, 1]) = [0, 1] \times \mathbb{R}$ is not compact.
If $X$ is compact and $Y$ is Hausdorff, every continuous $f: X \to Y$ is proper.

RemarkOpen Maps in Functional Analysis

The open mapping theorem in functional analysis states that a surjective bounded linear operator between Banach spaces is an open map. This is a deep result connecting the algebraic structure (linearity) with the topological structure (openness) and has no analogue for general topological spaces.