Section 2: Problem 4 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $f:A\rightarrow B$ and $g:B\rightarrow C$ .

(a) If $C_{0}\subset C$ , show that $(g\circ f)^{-1}(C_{0})=f^{-1}(g^{-1}(C_{0}))$ .

(b) If $f$ and $g$ are injective, show that $g\circ f$ is injective.

(d) If $f$ and $g$ are surjective, show that $g\circ f$ is surjective.

(e) If $g\circ f$ is surjective, what can you say about surjectivity of $f$ and $g$ ?

(f) Summarize your answers to (b)-(e) in the form of a theorem.

(a) $x\in(g\circ f)^{-1}(C_{0})\Leftrightarrow$ $g(f(x))\in C_{0}\Leftrightarrow$ $f(x)\in g^{-1}(C_{0})\Leftrightarrow$ $x\in f^{-1}(g^{-1}(C_{0}))$ .

(b)-(f) $g\circ f$ is injective iff $f$ is injective and $g$ is injective on $f(A)$ . It is surjective iff $g$ is surjective on $f(A)$ (and, therefore, is surjective on the larger set $B$ as well). In particular, if $f$ and $g$ are surjective, then $f(A)=B$ , and $g$ is surjective on $f(A)=B$ , i.e. $g\circ f$ is surjective.

Subpages

Section 2: Problem 4 Solution

Section 2: Problem 4 Solution