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### 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 ...
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### 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 $S$...
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### 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 ...
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### 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 ...
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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 ...
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 ...