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I've been thinking about this problem for a long time right now, and feel stuck.

Given that $X$ is a topological space, and that for $f$ to be lower semicontinuous, for any $x \in X$ and $\epsilon > 0$, there is a neighborhood of $x$ such that $f(x) - f(x') < \epsilon$ for all $x'$ in the neighborhood of $x$. Also, for the compact definition, I can only think of using every open cover of $K$ has a finite subcover for the topological definition of compact sets.

I don't know where I went wrong, but I don't see where I used the fact of $K$ being compact. So I'll put what I did.

Suppose that $\inf f(K)$ exists, then let's call it $a \in \mathbb{R}$. By the intermediate value theorem, we know that there exists some $x_0 \in X$ such that $f(x_0) = a$. So given that $f$ is lower-semicontinuous, then for $x_0 \in X$ and $\epsilon > 0$, there is a neighborhood of $x_0$ such that $f(x_0) - f(x') < \epsilon$ for all $x'$ in the neighborhood of $x_0$. Note there is an open set in the neighborhood of $x_0$ that contains $x_0$. Then since $K$ is compact, there is a finite open subcover of $K$. So call this finite subcover of $K$, $J$ and call the open set that contains $x_0$, $L$. Hence, $J \cup L$ is an open cover of $K$ and $f$ attains a minimum on $K$.

If the infimum of $f(K)$ does not exist, then we can't consider the case above. Take any $c \in f(K)$. Consider the set of the form $f_c := \{x \in X | f(x) \leq c\}$. Then we see each $f_c$ is closed since $f$ being lower-semicontinuous implies $f^{-1}(-\infty,a]$ is closed for any $a \in \mathbb{R}$ and $f(K)$ being unbounded. For all $c \in f(K)$, we have that the intersection of all $f_c$ gives us that the intersection of the $f_c$ is closed. Not sure if this fact is in contrary of $K$ being compact. I got stuck here.

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  • $\begingroup$ Notice that lower semicontinuity is equivalent to $f(x)\leq \liminf_{y\to x} f(y)$ for all $x$. Take a minimizing sequence, use compactness and conclude. $\endgroup$ – Jose27 Mar 23 '12 at 6:46
  • $\begingroup$ Hi Jose, the thing is i'm not allowed to use any other characterization of lower semicontinuity. But I'll take a look at it anyways for the heck of it. $\endgroup$ – MathNewbie Mar 23 '12 at 7:03
  • $\begingroup$ @MathNewbie: Why aren't you allowed to use any characterization of lower semiconinuity? $\endgroup$ – Beni Bogosel Mar 23 '12 at 7:04
  • $\begingroup$ I was informed by my instructor to strictly use the original definition I have, but I suppose that if I am to use the alternative characterization, it shouldn't be a problem given that I give a proof of their equivalence. At this point, I'll even try the alternative characterization of lower semicontinuity, since the definition that I've been given hasn't served me well or it's just me. $\endgroup$ – MathNewbie Mar 23 '12 at 7:10
  • $\begingroup$ Do you have that compact subsets of $\mathbb{R}$ are closed and bounded, and that bounded sequences have convergent subsequences? $\endgroup$ – Patrick Mar 23 '12 at 7:18
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Lower semicontinuity need not imply intermediate value property(IVP), and a function on a compact interval in $\mathbb{R}$ which satisfies IVP can fail to have the minimum.

But your $\inf = -\infty$ case proof can be elaborated to conclude that $f$ cannot be unbounded below. What you need is the finite intersection property of a compact set. Or, you can just consider the open cover (why?) formed by the open sets $U_c = \{ x : f(x) > c \}$ to obtain the boundedness of $f$.

Here is another possible approach:

Suppose $f$ has no minimum, and let $\alpha = \inf f(K)$. Then for each $x \in K$, we have $\alpha < f(x)$ and there is $\epsilon(x) > 0$ satisfying $$\alpha < f(x) - \epsilon(x).$$ Then by lower-semicontinuity, there exists a neighborhood $U(x)$ of $x$ such that $f(x) - \epsilon(x) < f(y)$ for all $y \in U(x)$. Now $U(x)$ covers $K$, so there are finitely many $x_1, \cdots, x_n \in K$ where $$K \subset U(x_1) \cup \cdots \cup U(x_n).$$ Let $\beta$ be given by $$ \beta = \min \{ f(x_k) - \epsilon(x_k) : k = 1, \cdots, n \}.$$ Then clearly $\alpha < \beta$, and for any $y \in K$, $y \in U(x_k)$ for some $k$ and hence $\beta \leq f(x_k) - \epsilon(k) < f(y)$, which means that $\beta \leq \inf f(K) = \alpha$, a contradiction! Therefore $f$ must attain its minimum at some point in $K$.

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So this won't help the person who needed it for homework, but for future visitors, here's an approach which is similar to the one above, but uses a contradiction of compactness.

Suppose $f$ has no minimum in $K$, then $\neg(\exists x\in K)(\forall y\in K)(f(x)\leq f(y))$ which is equivalent to $(\forall x\in K)(\exists y\in K)(f(y)<f(x))$ which is equivalent to $(\forall x\in K)(\exists y\in K)(x\in f^{-1}((f(y), +\infty)))$.

But this means the collection of open sets $\{f^{-1}((f(y), +\infty)):y\in K\}$ is an open cover of $K$.

We claim this open cover has no finite subcover.

Suppose $\{y_1, \cdots, y_n\}\subset K$ is finite, and define $N = \displaystyle\arg\min_{1\leq k\leq n}f(y_k)$, then $\displaystyle\bigcup_{1\leq k\leq n}(f(y_k),+\infty) = (f(y_N), +\infty)$. Then we have $\displaystyle y_N\in K$, but $y_N \notin f^{-1}((f(y_N), +\infty))= f^{-1}(\bigcup_{1\leq k\leq n}(f(y_k),+\infty)) = \bigcup_{1\leq k\leq n}f^{-1}((f(y_k),+\infty))$ and so $\{f^{-1}((f(y_k),+\infty)):1\leq k \leq n\}$ doesn't cover $K$.

This contradicts the compactness of $K$.

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  • $\begingroup$ Thank you for posting this answer @user1447786! I need a quick confirmation though -- {$y \in K$} can serve as the index for the open cover $f^{-1}(f(y), + \inf)$ despite the fact that it is uncountable, right? $\endgroup$ – Ye Tian Sep 13 '19 at 14:23
  • $\begingroup$ And where is the condition of lower semi-continuity applied? $\endgroup$ – Ye Tian Sep 13 '19 at 15:29
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    $\begingroup$ (1) Yup. Any open cover must have a finite subcover in a compact space, even if the cover has uncountably many open sets. (2) The fact that $f^{-1}(f(y),+\inf)$ is an open set for each $y\in K$ follows from the lower semi-continuity of $f$. $\endgroup$ – user1447786 Sep 16 '19 at 12:02
  • $\begingroup$ Thank you very much @@user1447786! $\endgroup$ – Ye Tian Sep 18 '19 at 0:47

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