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

302 Chap. 9 • EUCLIDEAN MANIFOLDS
A point x ∈E is a limit point of a subset U ⊂ E if every neighborhood of x contains a
point of U distinct from x . Note that x need not be in U . For example, the sphere
{ x d( x, x
0)
= ε}
are limit points of the open ball B( x
0, ε ). The closure of U ⊂ E , written U , is
the union of U and its limit points. For example, the closure of the open ball B( x
0, ε ) is the
closed ball B( x
0, ε ). It is a fact that the closure of U is the smallest closed set containing U .
Thus U is closed if and only if U = U .
The reader is cautioned not to confuse the concepts limit of a sequence and limit point of a
subset. A sequence is not a subset of E ; it is a function with values in E . A sequence may have a
limit when it has no limit point. Likewise the set of pints which represent the values of a sequence
may have a limit point when the sequence does not converge to a limit. However, these two
concepts are related by the following result from the theory of metric spaces: a point x is a limit
point of a set U if and only if there exists a convergent sequence of distinct points of U with x as
a limit.
A mapping f : U →E ', where U is an open set in E and E ' is a Euclidean point space or
an inner product space, is continuous at x ∈ 0
U if for every real number ε > 0 there exists a real
number δ ( x
0, ε ) > 0 such that d( xx ,
0) < δ ( ε, x
0)
implies d'( f( x0), f( x )) < ε . Here d ' is the
distance function for E ' . When f is continuous at x
0
, it is conventional to write
lim f( x) or f( x) → f( x ) as x→x
x→x0
0 0
The mapping f is continuous on U if it is continuous at every point of U . A continuous mapping
is called a homomorphism if it is one-to-one and if its inverse is also continuous. What we have
just defined is sometimes called a homeomorphism into. If a homomorphism is also onto, then it is
called specifically a homeomorphism onto. It is easily verified that a composition of two
continuous maps is a continuous map and the composition of two homomorphisms is a
homomorphism.
A mapping f : U →E ' , where U and E ' are defined as before, is differentiable at
there exists a linear transformation A ∈LV ( ; V ') such that
x
x ∈U if
f( x+ v) = f( x) + A v+
o( x, v )
(43.12)
x