Linked Questions

9
votes
2answers
5k views

Any lower semicontinuous function $f: X \to \mathbb{R}$ on a compact set $K \subseteq X$ attains a min on $K$.

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 &...
-1
votes
3answers
725 views

Prove that there exists a point $a$ in $A$ such that $| c-a | =\inf \{| c-x |: x \in A \}$?

Let $A$ be a nonempty compact subset of $\mathbb{R}$ and $c \in \mathbb{R}$. Prove that there exists a point $a$ in $A$ such that $| c-a | =\inf \{| c-x |: x \in A \}$?
3
votes
2answers
182 views

Path connected compact set with given property

Problem: Suppose $K$ is a compact subset of $\Bbb R^n$, and that for all $k_1, k_2 \in K$, there exists a continuous function $p:[0,1] \rightarrow K$ such that $p(0) = k_1 $ and $p_1 = k_2$. Then let $...
3
votes
3answers
137 views

Existence of such points in compact and connected topological space $X$

Let $X$ be a topological space which is compact and connected. $f$ is a continuous function such that; $f : X \to \mathbb{C}-\{0\}$. Explain why there exists two points $x_0$ and $x_1$ in $X$ such ...
0
votes
1answer
408 views

If $S$ is compact, show that there exists a point $x\in S$ such that $d(x,T)=d(S,T)$

$S,T\subset X$. $S$ is compact. $d(S,T):=inf\{d(s,t): s\in S, t\in T\}$. My idea is to look at the set $A:=\{d(s,T), s\in S\}$, where by definition $d(s,T):=inf\{d(s,t),t\in T\}$ I know that $S$...
0
votes
5answers
263 views

Find absolute maximum and minimum of the function on the given domain.

Given $f(x,y)=2x^4-xy^2+2y^2,0\le x\le 4, 0\le y\le2$. Find absolute extrema of $f(x,y)$. I have found $\partial f/\partial x=8x^3-y^2, \partial f/\partial y=-2xy+4y$ and after solving the equation ...
5
votes
5answers
72 views

Given $x_{0}$, let $f(x) = \|x-x_{0}\|$. Show tha $f$ has a minumum on any closed, nonempty set $A \subset \mathbb{R^{n}}$

Given $x_{0}$, let $f(x) = \|x-x_{0}\|$. Show tha $f$ has a minumum on any closed, nonempty set $A \subset \mathbb{R^{n}}$. I tried to do a test for reduction to the absurd, but it was a little ...
2
votes
2answers
39 views

about Euclidean Space

Suppose $E_1,E_2 \subseteq \Bbb R^m$ are closed sets and at least one of them is a bounded set. Prove that there exist $x_0\in E_1,y_0\in E_2$,such that $\rho (x_0,y_0)=\rho(E_1,E_2)$ Attempt :I ...
0
votes
0answers
42 views

Continuous Real Valueed Function

I need help to show that a set E of real numbers is closed and bounded if and only if every continuous function on E takes a maximum value. I can use The Intermediate Value Theorem to prove if E is ...
0
votes
4answers
28 views

A compact subset question

If $B$ is a compact subset of a metric space, how do you show that there exist points $x$ and $y$ such that the distance between them is equal to the diameter of $B$. Don’t know how to proceed. Any ...