Pasting Lemma

The pasting lemma is a fundamental result that allows us to construct continuous functions by defining them piecewise on closed (or open) subsets of the domain. It is used constantly in algebraic topology for constructing homotopies, paths, and maps on CW complexes.

Closed Pasting Lemma

Theorem2.6Pasting Lemma (Closed Version)

Let $X$ be a topological space with $X = A \cup B$ where $A$ and $B$ are closed subsets of $X$ . Let $f: A \to Y$ and $g: B \to Y$ be continuous functions such that $f(x) = g(x)$ for all $x \in A \cap B$ . Then the function $h: X \to Y$ defined by $h(x) = \begin{cases} f(x) & \text{if } x \in A, \\ g(x) & \text{if } x \in B \end{cases}$ is continuous.

Proof

Since $f$ and $g$ agree on $A \cap B$ , the function $h$ is well-defined.

Let $C \subseteq Y$ be closed. We show $h^{-1}(C)$ is closed in $X$ .

We have $h^{-1}(C) = f^{-1}(C) \cup g^{-1}(C)$ , where we interpret $f^{-1}(C) \subseteq A$ and $g^{-1}(C) \subseteq B$ .

Since $f: A \to Y$ is continuous (in the subspace topology on $A$ ), $f^{-1}(C)$ is closed in $A$ . Since $A$ is closed in $X$ , the set $f^{-1}(C)$ is closed in $X$ .

Similarly, $g^{-1}(C)$ is closed in $B$ , and since $B$ is closed in $X$ , $g^{-1}(C)$ is closed in $X$ .

Therefore $h^{-1}(C) = f^{-1}(C) \cup g^{-1}(C)$ is a finite union of closed sets, hence closed in $X$ .

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Open Pasting Lemma

Theorem2.7Pasting Lemma (Open Version)

Let $X = \bigcup_{\alpha \in A} U_\alpha$ where each $U_\alpha$ is open in $X$ . Let $f_\alpha: U_\alpha \to Y$ be continuous functions such that $f_\alpha = f_\beta$ on $U_\alpha \cap U_\beta$ for all $\alpha, \beta$ . Then the function $f: X \to Y$ defined by $f(x) = f_\alpha(x)$ for $x \in U_\alpha$ is well-defined and continuous.

Proof

Well-definedness follows from the compatibility condition. For continuity, let $V \subseteq Y$ be open. Then: $f^{-1}(V) = \bigcup_{\alpha \in A} f_\alpha^{-1}(V).$

Each $f_\alpha^{-1}(V)$ is open in $U_\alpha$ (since $f_\alpha$ is continuous), hence open in $X$ (since $U_\alpha$ is open in $X$ ). Therefore $f^{-1}(V)$ is a union of open sets, hence open.

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RemarkClosed Version Requires Finiteness

The closed pasting lemma generalizes to finitely many closed sets $X = C_1 \cup \cdots \cup C_n$ . However, it does not extend to arbitrary families of closed sets. For example, consider $X = [0, 1]$ and $C_x = \{x\}$ for each $x \in [0, 1]$ . Define $f_x: \{x\} \to \mathbb{R}$ by $f_x(x) = \chi_{\mathbb{Q}}(x)$ . Each $f_x$ is trivially continuous, but the pasted function is the characteristic function of the rationals, which is nowhere continuous.

The open version has no such restriction because arbitrary unions of open sets are open.

Applications

ExampleConcatenation of Paths

Let $\gamma_1: [0, 1] \to X$ and $\gamma_2: [0, 1] \to X$ be continuous paths with $\gamma_1(1) = \gamma_2(0)$ . The concatenation $\gamma_1 * \gamma_2: [0, 1] \to X$ is defined by: $(\gamma_1 * \gamma_2)(t) = \begin{cases} \gamma_1(2t) & \text{if } t \in [0, 1/2], \\ \gamma_2(2t - 1) & \text{if } t \in [1/2, 1]. \end{cases}$

The two pieces are continuous (compositions of continuous maps), defined on the closed sets $[0, 1/2]$ and $[1/2, 1]$ , and agree at $t = 1/2$ : $\gamma_1(2 \cdot 1/2) = \gamma_1(1) = \gamma_2(0) = \gamma_2(2 \cdot 1/2 - 1).$

By the pasting lemma, $\gamma_1 * \gamma_2$ is continuous.

ExampleConstruction of Homotopies

Many homotopy constructions rely on the pasting lemma. For instance, to show that path homotopy is transitive: if $F: \gamma_0 \simeq \gamma_1$ and $G: \gamma_1 \simeq \gamma_2$ are path homotopies, define $H: [0,1] \times [0,1] \to X$ by: $H(s, t) = \begin{cases} F(s, 2t) & \text{if } t \in [0, 1/2], \\ G(s, 2t-1) & \text{if } t \in [1/2, 1]. \end{cases}$

The domain is divided into two closed rectangles $[0,1] \times [0, 1/2]$ and $[0,1] \times [1/2, 1]$ , the restrictions are continuous, and they agree on the intersection $[0,1] \times \{1/2\}$ where both give $\gamma_1(s)$ . By the pasting lemma, $H$ is a continuous homotopy $\gamma_0 \simeq \gamma_2$ .

ExampleMaps on CW Complexes

In the theory of CW complexes, one builds spaces inductively by attaching cells. At each stage, maps are defined on closed cells and pasted together. The pasting lemma (in its finite and locally finite versions) ensures continuity of the resulting global map.

More precisely, CW complexes carry the weak topology: $A \subseteq X$ is closed if and only if $A \cap \overline{e}$ is closed for every cell $e$ . This is a colimit topology that makes pasting automatic.