A set
of mappings from a set
into a
topological space
with some natural topology
on
.
For fixed
and
one obtains different spaces of mappings, depending on which mappings
are included in
and what natural topology
is endowed with. The choice of
is related to the presence of additional structures on
and
and to peculiarities of the situation considered. Thus, for
one can take: the set of all continuous mappings
,
the set of all mappings
,
the set of all continuous linear mappings from a topological vector space
into a topological vector space
,
the set of all continuous homomorphisms from a topological group
into a topological group
,
the set of all smooth mappings from an interval into the straight line, etc.
The importance of considering spaces of mappings is to a certain
extent related to the fact that mappings are
the most general method of comparing mathematical objects.
Natural topologies (on
)
are usually determined in the following way. A family
of subsets of
is fixed, and a
pre-base
for a topology
on
is formed by sets of the form
where
and
is an open set in
.
If
is the family of finite (or singleton) subsets of
,
then
is called the
topology of pointwise convergence
on
.
If
consists of all compact subsets of
,
then
is called the
compact-open topology.
If
,
then
is called the
topology of uniform convergence
(on
).
Moreover, every topology
on
obtained by this scheme is called the topology of uniform convergence on elements of
.
Depending on the branch of mathematics, some spaces of mappings turn
out to be especially important. Among the central objects in
functional analysis one counts the Banach spaces
of continuous functions on compacta in the
norm topology,
i.e. the topology of uniform convergence, and in the
weak topology,
which can be described in terms of pointwise convergence. In homotopy theory
an important role is played by the path space of
a topological space, i.e. the space of continuous mappings
from a closed interval into this topological space. Homotopy of one mapping into
another is represented by a path in the space of mappings. The space
of mappings from a sphere into a sphere arises
in the definition of homotopy and cohomotopy groups.
The compact-open topology on the set of mappings of one
-space
into another turns out to be especially natural. An advantage of
the topology of uniform convergence (on the entire space) is its
metrizability. This topology is the strongest in a large class of
natural topologies on a space of mappings. However, the topology of
pointwise convergence too has its advantages — as the weakest in
this class of topologies. First, this topology best reflects the compactness,
and compactness is one of the most useful properties of a set of functions. Secondly, there
is the fundamental result of
J. Nagata
that puts the study of arbitrary Tikhonov spaces (cf.
Tikhonov space)
in direct relation to the study of topological rings. More precisely, two Tikhonov spaces
are homeomorphic if and only if the topological rings
and
of continuous functions on
and
,
respectively, with the topology of pointwise convergence are topologically isomorphic.
The consideration of topological properties of spaces of mappings is useful
in proving theorems on the existence of mappings with some
property. The completeness of the metric space of continuous real-valued
functions on a compactum is used, via the contraction-mapping principle,
in the proof of the fundamental theorem on the existence
of a solution to a differential equation under certain assumptions.
A consequence of the completeness of a metric space of functions is the
Baire property.
Using it one can prove, e.g., the existence of a continuous nowhere-differentiable
function on an interval. The Baire property of spaces of functions
plays a central role in the proof of the theorem on
general position,
in the proof of the well-known theorem on the imbeddability of each
-dimensional
compactum with a countable base into
-dimensional
Euclidean space, etc.
The influence of spaces of real-valued functions on general topology becomes
clear in the following problem, of a general character:
In what way are the properties of two spaces
and
related if the spaces of continuous real-valued functions on them (in
the topology of pointwise convergence, in the compact-open topology) are
homeomorphic (linearly homeomorphic). It is known, e.g.,
that linear homeomorphisms preserve compactness and dimension.
The fact that duality between properties of a topological space and
topological properties of the space of functions on them with the
topology of pointwise convergence is inherited is of special significance. As an example
of a useful result in this respect one can invoke the following theorem:
Any finite power of a space is Lindelöf if and only if the
space of functions on it has countable tightness. This result is
used, in particular, in the study of the structure of
Eberlein compacta
— compacta in Banach spaces, endowed with the weak topology.