Decide whether of the following sets are compact? Justify your answers. i. {(x,

Question

Decide whether of the following sets are compact? Justify your
answers.

i. {(x, y) R2 : 2x2 y2 1}.

ii. {(ex cos , ex sin ) : x 0, 0 2}.

iii. (0, 1] in R.

Explanation / Answer

http://www.mathcs.org/analysis/reals/topo/proofs/cpctbdd.html By above link A set S of real numbers is compact if and only if it is closed and bounded. Here are some theorems that can be used to shorten proofs that a set is open or closed. A union of open sets is open, as is an intersection of finitely many open sets. An intersection of closed sets is closed, as is a union of finitely many closed sets. If X ightarrow Y is a continuous function and Zsubset Y is open/closed, then f^{-1}(Z) is open/closed. Let R ightarrowR be a continuous function. Then the graph of f is closed. A direct proof of this would be to take some point (x,y) with y e f(x) and argue that there exists delta>0 such that if (x',y') has distance at most delta from (x,y) then y' e f(x'). That can be done, but it is slightly tedious. A quick proof is to consider the map (x,y) ightarrow y-f(x). This is continuous, and the graph of f is F^{-1}({0}). As {0} is closed.Therefore, the graph is closed. i) 2x^2 - y^2 = 1 x^2/(1/2)-y^2=1 is equation of hyperbola Hence set of elements which satisfy 2x^2 - y^2 = 1 is not bounded.Thus, ,it is not compact as a subset of set is not bounded ii.{(e^-x cos ?, e^-x sin ?) : x = 0, 0 = ? = 2p}=(z,y) pts satisfy z^2+y^2= e^-2x for some x All pts in R^2 except pts which satisfy z^2+y^2

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