Continuity and Homeomorphisms

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.

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 $x = 0$.