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Suppose $Y$ be an ordered set in the order topology. Let $f, g: X \to Y$ be continuous. How to show that the set $\{x| f(x) \leq g(x)\}$ is closed?

This is a excercise from munkres. Maybe trying to show that the set $f(\{x| f(x) > g(x)\})$ is open is suffice. But I could not figure out the connection between this set and the continuity of the two functions.Could you give a hint?

Thanks.

2 Answers 2

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The set $$U = \{(y_1, y_2) \in Y \times Y \mid\, y_1 \leq y_2 \,\}$$ is closed in $Y \times Y\,$:

Suppose $(z_1, z_2) \in Y \times Y\,$ is not in it, so that $z_1 > z_2$.

If there is some $z_3 \in Y$ such that $z_1 > z_3 > z_2$, the basic open set $(z_3, \rightarrow) \times (\leftarrow, z_3)$ contains $(z_1, z_2)$ and misses $U$, otherwise $z_1$ and $z_2$ are neighbours and $(z_2, \rightarrow) \times (\leftarrow, z_1)$ has this property. Note that all used subsets in $Y$ are open in the order topology.

Now $f \nabla g: X \rightarrow Y \times Y$ is continuous (standard in the product topology) when $f$ and $g$ are, and your set is exactly

$$(f \times g)^{-1}[U]$$

and thus closed as the inverse image of a closed set under a continuous function.

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    Calling a closed set $U$ is strange :-) But I guess it was $U$ for *upper* half space...2011-09-21
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    @lhf: it was for "upper" indeed.2011-09-21
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    @HennoBrandsma I have just editted a post of yours from 2011! Do you agree to this edit? I guess you mean the map $f \times g \colon X \rightarrow Y \times Y$ defined by $\left( f \times g \right) (x) \colon= f(x) \times g(x)$ for all $x \in X$. Am I right?2017-09-19
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    @SaaqibMahmuud yes, you're almost right. I meant the diagonal product $f \nabla g$ that sends $x$ to $(f(x),g(x))$ of course.2017-09-19
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    @HennoBrandsma thanks. What do you mean by "almost right"?2017-09-19
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    But where does the existence of $z_3$ comes from ?2017-12-18
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    @onurcanbektas there are 2 cases: such a $z_3$ exists or not2017-12-18
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    @HennoBrandsma I read your proof in the morning, so being little sleepy led me to not being able to see that "if" part :). Thanks for your response.2017-12-18
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    @SaaqibMahmuud I'm against the confusing notation $f(x) \times g(x)$ for $(f(x), g(x))$.2017-12-19
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The complement of $\{x : f(x) \leq g(x)\}$ is $\{x : f(x) > g(x)\}$. Part of the latter set is $$\bigcup_{y\in Y} \{x : f(x) > y > g(x)\},$$ which is open because each set is the intersection of two open sets. What is missing?