MHB Does Compactness Ensure a Positive Minimum for Continuous Functions?

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Let $K \subset \mathbb{R^n}$ be compact and let $f: K \rightarrow \mathbb{R}$ be continuous. Suppose that $f(x) > 0$ $\forall x \in S.$ Prove there is a $c > 0$ such that $f(x) \geq c$ $\forall x \in K$

My Sol:

I said that by the extreme value theorem $\exists a,b \in K $ such that $f(a) \leq f(x) \leq f(b) \forall x\in K$ so if we let $c=f(a) $ then $c> 0$ and $f(x) \geq c$ a number c > 0 such that f(x) ≥ c for every x ∈ K
 
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Hi kalvin,

Assuming $S = K$, your solution is correct.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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