MAT441 Homework Assignment 4
1. Let A:= ©(0, y) : 1
2≤y≤1ªand B:= ©(x, y) : sin ¡1
x¢,0< x ≤1ª. (See Figure 1.)
Show that every point of Ais a limit point of B. This implies that the sets Aand Bare not
separated and that A∪Bis a connected set as we discussed in class.
0.5
0.2
0
-0.5
0
-1
10.80.6
1
0.4
Figure 1: The sets Aand Bin problem 1
2. Let A:= ©(x, y) : y=x
n, n ∈Z+ªand B:= ©(x, 0) : 1
2≤x≤1ª. (See Figure 2.)
0.2
0.6
0 0.8
0.4
1
0.6
0.2
0
0.4
0.8
1
Figure 2: The sets Aand Bin problem 2
(a) Show that A∪Bis connected.
(b) Show that A∪Bis closed.
