Introduction to Vectors and Tensors Vol 2 (Bowen 246). - Index of

300 Chap. 9 • EUCLIDEAN MANIFOLDS
{ d ε}
B( x , ε ) = x ( x, x ) ≤
(43.11)
0 0
A neighborhood of x ∈E is a set which contains an open ball centered at x . A subset U of E is
open if it is a neighborhood of each of its points. The empty set ∅ is trivially open because it
contains no points. It also follows from the definitions that E is open.
Theorem 43.2. An open ball is an open set.
Proof. Consider the open ball B( x
0, ε ). Let x be an arbitrary point in B( x
0, ε ). Then
ε − d( xx ,
0) > 0 and the open ball B( x, ε − d( x, x
0))
is in B( x
0, ε ), because if
y ∈B( x, ε −d( x, x
0))
, then d( y, x) < ε −d( x0, x ) and, by (43.8), d( x0, y) ≤ d( x0, x) + d( x, y ),
which yields d( x0, y ) < ε and, thus, y∈ B( x
0, ε ).
A subset U of E is closed if its complement, EU, is open. It can be shown that closed
balls are indeed closed sets. The empty set, ∅ , is closed because E = E ∅ is open. By the same
logic E is closed since EE =∅ is open. In fact ∅ and E are the only subsets of E which are
both open and closed. A subset U ⊂ E is bounded if it is contained in some open ball. A subset
U ⊂ E is compact if it is closed and bounded.
Theorem 43.3. The union of any collection of open sets is open.
Proof. Let { U
α
α ∈ I}
be a collection of open sets, where I is an index set. Assume that
x ∈∪
α∈IUα
. Then x must belong to at least one of the sets in the collection, say U
α 0
. Since U
α 0
is
open, there exists an open ball B( x, ε ) ⊂Uα ⊂∪ 0 α∈IUα
. Thus, B( x, ε ) ⊂ ∪
α∈IUα
. Since x is
arbitrary, ∪ is open.
U α∈I
α
Theorem43.4. The intersection of a finite collection of open sets is open.
Proof. Let { }
1 ,..., α
n
U U be a finite family of open sets. If ∩ U i= 1 i
is empty, the assertion is trivial.
Thus, assume ∩ n
U is not empty and let x be an arbitrary element of ∩ n
U . Then i= 1 i
i= 1 i
x ∈U i
for
i = 1,..., n and there is an open ball B( x, ε i
) ⊂U i
for i = 1,..., n. Let ε be the smallest of the
positive numbers ,..., n
n
ε1 ε
n
. Then x∈B( x, ε ) ⊂∩ i=
1U
i
. Thus ∩ U i= 1 i
is open.