ASSIGNMENT 7
MAT 472 ·FALL 2005
Also look at: Problems 26–38 from Chapter III of Rosenlicht.
Problem 1 (See Problem III.31).Prove directly that [0,1] is a compact subset of Ras
follows: Given an open cover {Uα|α∈I}of [0,1], define
S={x∈(0,1] |[0, x]⊆ ∪α∈FUαfor some finite set F⊆I},
and show that sup S∈Sand sup S= 1.
Problem 2 (Problem III.32).Prove that the union of a finite number of compact sets in
any metric space is compact.
Problem 3 (Problem III.33).Let Sbe a sequentially compact set in a metric space (E , d),
and let Cbe an open cover of S.Without assuming that Sis compact, show that there exists
> 0 such that for each p∈S, there exists U∈ C such that B(p)⊆U.
Problem 4. Let Sbe a sequentially compact set in a metric space (E , d). Use the result of
Problem 3 and the fact that every sequentially compact set is totally bounded to prove that
Sis compact.
Date: October 4, 2005 / Due Date: Tuesday, October 11, 2005.
S. Kaliszewski, Department of Mathematics and Statistics, Arizona State University.
1
