Answer in Real Analysis for Dumela #211190

Question #211190

Let f : M → Y be a homeomorphism and O an open subset of M. Explain

concisely in no more than two lines of text why f(O) is an open set

Expert's answer

Since $f(O)=(f^{-1})^{-1}(O)$ is a preimage of an open subset $O\subset M$ under a continuous map $f^{-1}:Y\to M,$ we conclude that $f(O)$ is also open.

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Assignment Expert

15.07.21, 23:45

Dear Matimu Dumela, please use the panel for submitting a new question.

Matimu Dumela

29.06.21, 09:09

Let A = [ [2, 7) ∪ Z ] ∩ (−1, 4) be equipped with the induced metric from R. Let f : A → R be any given map. (a) Give an explicit and simplified expression for A. (b) Give explicitly the set S of all points of A at which f is continuous.

Matimu Dumela

29.06.21, 09:08

Let Wj ⊂ M for all j = 1, . . . , p, and x0 ∈ W = p ∩ j=1 Wj . Let λj > 0 be such that B(x0, λj ) ⊂ Wj for all j. Give an -neighborhood V of x0 contained in W.

Matimu Dumela

29.06.21, 09:06

Assume that the metric space M satisfies the following properties. (P1) For all δ > 0, M is contained in the union of a finite number of open balls of radius δ. (P2) For every open cover (Eα)α∈I of M, there is some δ > 0 such that ∀x ∈ M, ∃α ∈ I such that B(x, δ) ⊂ Eα. Show clearly and concisely that M is compact, using the Heine-Borel property.