Continuity in Topological Spaces

Continuity is a cornerstone of mathematical analysis, but its essence extends far beyond calculus. In topology, continuity is redefined through the elegant lens of open sets, providing a unified framework that applies to diverse spaces from manifolds to function spaces. Mastering this generalization unlocks tools like homeomorphisms and embeddings, which are essential for understanding the shape and structure of mathematical objects.

Defining Continuity via Preimages of Open Sets

In metric spaces, you define continuity using epsilon-delta arguments that rely on distance. Topological spaces discard metrics, so continuity must be captured more fundamentally. The key insight is that a continuous map between topological spaces preserves the "closeness" encoded by open sets through preimages. Formally, a function $f:X\to Y$ between topological spaces 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 $X$. This means that pulling back open sets through $f$ always yields open sets.

This definition generalizes the epsilon-delta approach: in a metric space, open balls form a basis, and the preimage condition ensures that points near $f(x)$ come from points near $x$. For example, consider the identity map $id:\mathbb{R}\to \mathbb{R}$ where both copies of $\mathbb{R}$ have the usual topology. Since any open interval's preimage is itself, $id$ is continuous. Conversely, a function like the Dirichlet function—which is 1 on rationals and 0 on irrationals—is discontinuous because the preimage of $\{1\}$ is the rationals, which is not open in the standard topology. This preimage-based definition seamlessly extends continuity to settings without a clear distance, such as the Zariski topology in algebraic geometry or function spaces equipped with the topology of pointwise convergence.

Homeomorphisms: Topological Equivalences

When two topological spaces are essentially the same in shape, we formalize this with homeomorphisms. A homeomorphism is a bijective continuous map $f:X\to Y$ whose inverse $f^{-1}:Y\to X$ is also continuous. Such a map establishes a topological equivalence, meaning $X$ and $Y$ have identical topological properties—like connectedness, compactness, or the number of holes—even if they look different geometrically.

Homeomorphisms act as "isomorphisms" in topology. For instance, a circle and a square are homeomorphic in the plane because you can continuously deform one into the other without cutting or gluing; a explicit homeomorphism might stretch and bend the square into the circle. In contrast, a circle and a figure-eight are not homeomorphic because removing a single point from a circle leaves it connected, whereas removing the intersection point from a figure-eight disconnects it. This invariance under homeomorphisms is why topologists often say a coffee mug and a doughnut are the same: both are surfaces of genus one (one hole), and there exists a continuous deformation between them. Recognizing homeomorphisms allows you to classify spaces and understand which properties are truly topological versus geometric.

Open and Closed Maps

Continuity concerns preimages, but it's also useful to examine how maps handle images of sets. An open map is a function $f:X\to Y$ such that for every open set $U\subseteq X$, the image $f(U)$ is open in $Y$. Similarly, a closed map sends closed sets to closed sets. These concepts are independent of continuity: a map can be open but not continuous, or continuous but not open.

For example, consider the projection $\pi :{\mathbb{R}}^2\to \mathbb{R}$ defined by $\pi (x,y)=x$. This is an open map because the image of any open disk in the plane is an open interval on the real line. However, $\pi$ is continuous as well, as the preimage of an open interval is an infinite strip open in ${\mathbb{R}}^2$. As a non-example, the map $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=x^2$ is continuous but not open: the open interval $(-1,1)$ is mapped to $[0,1)$, which is not open. Homeomorphisms are special because they are necessarily both open and closed, reflecting the perfect correspondence between topologies. In analysis, open maps are crucial in theorems like the Open Mapping Theorem for complex analytic functions, which states that non-constant analytic functions are open maps.

Topological Embeddings

Sometimes, you want to realize one space as a subspace of another without altering its intrinsic topology. A topological embedding is an injective continuous map $f:X\to Y$ that is a homeomorphism onto its image $f(X)\subseteq Y$, where $f(X)$ carries the subspace topology from $Y$. This means $f$ preserves all topological properties of $X$ within $Y$, allowing $X$ to be seen as a "copy" inside $Y$.

Embeddings are ubiquitous in geometry and analysis. In geometry, the torus (surface of a doughnut) can be embedded in ${\mathbb{R}}^3$ as a familiar ring, but also in ${\mathbb{R}}^4$ in ways that avoid self-intersection. In analysis, consider the embedding of the real line $\mathbb{R}$ into the plane ${\mathbb{R}}^2$ via $f(t)=(t,\sin t)$, which traces a sine curve. This map is injective and continuous, and the curve with the subspace topology is homeomorphic to $\mathbb{R}$, so it's an embedding. However, not every injective continuous map is an embedding. For instance, map the interval $[0,2\pi )$ onto the unit circle in ${\mathbb{R}}^2$ by $t\mapsto (\cos t,\sin t)$. This is injective and continuous, but its image—the circle—is compact, while $[0,2\pi )$ is not compact in the standard topology. The inverse map from the circle back to $[0,2\pi )$ is not continuous at the point corresponding to $0$ and $2\pi$, so this is not an embedding. Embeddings ensure that the subspace topology matches exactly, which is vital for studying manifolds or function spaces where you want to preserve local structure.

Common Pitfalls

1. Confusing continuity with openness: A continuous map need not send open sets to open sets. For example, the constant map $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=5$ is continuous because the preimage of any set containing 5 is $\mathbb{R}$ (open) and others are empty (open). But the image of an open set is $\{5\}$, which is not open in the standard topology. Remember: continuity is about preimages, while openness is about images.

1. Assuming homeomorphisms are just bijective continuous maps: A bijective continuous map might not have a continuous inverse. Consider $X=[0,2\pi )$ with the standard topology and $Y$ as the unit circle with the subspace topology from ${\mathbb{R}}^2$. The map $f(t)=(\cos t,\sin t)$ is bijective and continuous, but $f^{-1}$ is not continuous at $(1,0)$ because small arcs around that point correspond to intervals near both $0$ and $2\pi$, breaking continuity. Always verify that the inverse map is continuous.

1. Overlooking the subspace topology in embeddings: An injective continuous map $f:X\to Y$ is an embedding only if it induces a homeomorphism with its image under the subspace topology. For instance, if you equip $X$ with a finer topology, an otherwise nice map might fail to be an embedding. Ensure that the topology on $f(X)$ inherited from $Y$ matches the original topology on $X$ via $f$.

1. Misapplying concepts to closed maps: Just because a map is closed does not imply it is continuous or open. For example, the floor function $\lfloor x\rfloor :\mathbb{R}\to \mathbb{Z}$ (with $\mathbb{Z}$ having the discrete topology) sends closed sets to closed sets but is not continuous because the preimage of a singleton integer is a half-open interval, which is not open in $\mathbb{R}$. Analyze each property—continuity, openness, closedness—independently unless theorems specify relationships.

Summary

• Continuity in topology is defined by the preimage of every open set being open, generalizing metric-space continuity without relying on distance.
• Homeomorphisms are bijective continuous maps with continuous inverses, serving as topological equivalences that preserve properties like connectedness and compactness.
• Open and closed maps focus on images of sets: open maps send open sets to open sets, while closed maps send closed sets to closed sets; these are separate from continuity.
• Topological embeddings are injective continuous maps that homeomorphically map a space onto its image, allowing one space to be realized as a subspace of another without distortion.
• Examples from geometry (e.g., tori in ${\mathbb{R}}^3$) and analysis (e.g., function space embeddings) illustrate how these concepts unify and apply across mathematics.