Section 18: Problem 8 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 $Y$ be an ordered set in the order topology. Let $f,g:X\rightarrow Y$ be continuous.

(a) Show that the set $\{x|f(x)\le g(x)\}$ is closed in $X$ .

(b) Let $h:X\rightarrow Y$ be the function $h(x)=min\{f(x),g(x)\}.$ Show that $h$ is continuous. [Hint: Use the pasting lemma.]

(a) $\{x|f(x)>g(x)\}=\cup_{y\in Y}\{x|f(x)>y>g(x)\}\cup$ $\cup_{y<y':(y,y')=\emptyset}\{x|f(x)>y,g(x)<y'\}$ is open as all sets on the right are intersections of two open sets (preimages of open rays).

(b) $h(x)=f(x)$ on $\{x|f(x)\le g(x)\}$ , and $h(x)=g(x)$ on $\{x|f(x)\ge g(x)\}$ , therefore, $h$ restricted to these sets is continuous, and both sets are closed by (a). Using the pasting lemma, $h$ is continuous.

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Section 18: Problem 8 Solution

Section 18: Problem 8 Solution