Continuity and Homeomorphisms | Mathematics

Definition. Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A function $f : X \to Y$ is continuous if the preimage of every open set is open: for all $U \in \tau_Y$, $f^{-1}(U) \in \tau_X$.

Equivalently, $f$ is continuous if the preimage of every closed set is closed.

Proposition 4.1. The following are equivalent for $f : X \to Y$:

1. $f$ is continuous.
2. $f^{-1}(V)$ is closed in $X$ for every closed $V \subseteq Y$.
3. $f^{-1}(\overline{B}) \subseteq \overline{f^{-1}(B)}$ for every $B \subseteq Y$.
4. $f(\overline{A}) \subseteq \overline{f(A)}$ for every $A \subseteq X$.

Example 4.1. Every constant function $f : X \to Y$ is continuous.

Example 4.2. The identity map $\operatorname{id} : X \to X$ is always continuous.

Example 4.3. If $\tau_1$ and $\tau_2$ are topologies on $X$ with $\tau_1 \subseteq \tau_2$, then $\operatorname{id} : (X, \tau_2) \to (X, \tau_1)$ is continuous, but $\operatorname{id} : (X, \tau_1) \to (X, \tau_2)$ need not be.

Proposition 4.2. The composition of continuous functions is continuous: if $f : X \to Y$ and $g : Y \to Z$ are continuous, then $g \circ f : X \to Z$ is continuous.

Continuity at a point. A function $f : X \to Y$ is continuous at $x_0 \in X$ if for every neighbourhood $V$ of $f(x_0)$, there exists a neighbourhood $U$ of $x_0$ such that $f(U) \subseteq V$. This local definition coincides with the global definition: $f$ is continuous iff it is continuous at every point.

Proposition 4.3 (Pasting Lemma). If $X = A \cup B$ where $A$ and $B$ are closed (or open) in $X$, and $f : A \to Y$, $g : B \to Y$ are continuous functions that agree on $A \cap B$, then the function $h : X \to Y$ defined by $h|_A = f$ and $h|_B = g$ is continuous.

Example 4.4 (Piecewise functions). The absolute value function $f(x) = |x|$ is continuous on $\mathbb{R}$ by the pasting lemma: on $(-\infty, 0]$ it equals $-x$ (continuous), on $[0, \infty)$ it equals $x$ (continuous), and the two agree at $x = 0$.

Example 4.5. The function $f(x) = \sin(1/x)$ for $x \neq 0$ and $f(0) = 0$ is not continuous at $0$, even though it satisfies the intermediate value property on every neighbourhood.

Definition. A function $f : X \to Y$ is open if the image of every open set is open; it is closed if the image of every closed set is closed. Continuity does not imply openness: the constant function $f(x) = c$ is continuous but not open (the image $\{c\}$ is not open in $\mathbb{R}$). The projection $\pi : X \times Y \to X$ is open but not necessarily closed.

Proposition 4.4. A bijective continuous function is a homeomorphism iff it is open (equivalently, closed).

Definition. A function $f : X \to Y$ is a homeomorphism if $f$ is bijective and both $f$ and $f^{-1}$ are continuous. We write $X \cong Y$ and say $X$ and $Y$ are homeomorphic.

A topological property (or topological invariant) is a property preserved by homeomorphisms.

Example 4.6. $(0, 1)$ is homeomorphic to $\mathbb{R}$ via $f(x) = \tan\left(\pi x - \frac{\pi}{2}\right)$.

Example 4.7. Any two open intervals $(a, b)$ and $(c, d)$ in $\mathbb{R}$ are homeomorphic via an affine map.

Example 4.8 (Coffee cup and donut). A coffee cup (with a handle) is homeomorphic to a torus $S^1 \times S^1$ via a continuous deformation. This illustrates that homeomorphisms permit stretching and bending but not tearing or gluing.

Proposition 4.5. Homeomorphism is an equivalence relation: reflexive, symmetric, and transitive.

The following are topological invariants (preserved by homeomorphisms):

• Compactness
• Connectedness
• Separation axioms ($T_0$, $T_1$, $T_2$, etc.)
• Countability axioms (first-countable, second-countable)
• The fundamental group $\pi_1(X)$

Proposition 4.6. “Boundedness” is not a topological property: $(0, 1)$ is bounded but is homeomorphic to the unbounded $\mathbb{R}$.

Definition. An embedding is a homeomorphism onto its image: $f : X \to Y$ such that $f : X \to f(X)$ (with the subspace topology) is a homeomorphism.

Definition. A surjective continuous map $q : X \to Y$ is a quotient map if $U \subseteq Y$ is open iff $q^{-1}(U)$ is open in $X$. Quotient maps are used to construct spaces by gluing: if $\sim$ is an equivalence relation on $X$, then $X/{\sim}$ with the quotient topology is the space of equivalence classes with the finest topology making the projection continuous.

Example 4.9. The unit interval $[0, 1]$ with endpoints identified yields the circle $S^1$: $[0, 1] / \{0 \sim 1\} \cong S^1$. The unit square with opposite edges identified yields the torus.

Problem. Show that $(0, 1)$ and $[0, 1]$ are not homeomorphic.

A homeomorphism is the topological version of “the same shape.” Two spaces are homeomorphic if there exists a continuous bijection between them whose inverse is also continuous. Informally, this means you can stretch, bend, and deform one space into the other, but you cannot tear or glue. A coffee cup and a donut are homeomorphic because they both have exactly one hole; a sphere and a torus are not homeomorphic because they have different numbers of holes.

The key insight is that homeomorphisms preserve topological properties such as compactness, connectedness, and the fundamental group. Properties that are not preserved by homeomorphisms, like “boundedness” or “being a subset of $\mathbb{R}^3$,” are geometric rather than topological. The pasting lemma captures a practical criterion for constructing continuous functions: if you define a function separately on two closed (or open) pieces that agree on the overlap, the result is continuous. This is why absolute value, piecewise linear functions, and most “glued together” constructions in analysis turn out to be continuous.

Problem. Determine whether $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x\sin(1/x)$ for $x \neq 0$ and $f(0) = 0$ is continuous at $x = 0$.

Mistake 1: Assuming that a continuous bijection is automatically a homeomorphism. A continuous bijection is not necessarily a homeomorphism. The inverse function must also be continuous. For example, the map $f: [0, 2\pi) \to S^1$ defined by $f(t) = (\cos t, \sin t)$ is a continuous bijection, but its inverse is not continuous at $(1, 0)$. A bijective continuous map is a homeomorphism if and only if it is open (or equivalently, closed).

Mistake 2: Confusing open maps with continuous maps. A function can be continuous without being open, and vice versa. The constant function $f(x) = c$ is continuous but not open (the image of any open set is $\{c\}$, which is not open). The projection $\pi: \mathbb{R}^2 \to \mathbb{R}$ is open but not closed (the image of the closed hyperbola $\{(x, y) : xy = 1\}$ is $\mathbb{R} \setminus \{0\}$, which is not closed).

Mistake 3: Forgetting that homeomorphism is an equivalence relation. Homeomorphism is reflexive (the identity map is a homeomorphism), symmetric (the inverse of a homeomorphism is a homeomorphism), and transitive (the composition of homeomorphisms is a homeomorphism). Failing to check any of these can lead to errors when classifying spaces up to homeomorphism.

Mistake 4: Assuming that topological properties are preserved by continuous maps. Only homeomorphisms preserve all topological properties. A continuous map need not preserve compactness, connectedness, or separation axioms in both directions. For example, a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, but this is a special case, not a general rule.

Mistake 5: Misapplying the pasting lemma. The pasting lemma requires that the two pieces agree on their intersection and that both pieces are either both closed or both open. If one piece is closed and the other is open, or if they disagree on the intersection, the resulting function may not be continuous.

flowchart TD
    A[4_Continuity And Homeomorphisms] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

• Topological Spaces: Topological spaces provide the foundation for defining continuity and homeomorphisms.
• Compactness: Compactness is preserved by continuous functions and characterizes spaces where extreme value theorem holds.
• Connectedness: Connectedness is a topological invariant preserved by homeomorphisms and continuous images.