Pointed space explained
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as
that remains unchanged during subsequent discussion, and is kept track of during all operations.
Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map
between a pointed space
with basepoint
and a pointed space
with basepoint
is a based map if it is continuous with respect to the topologies of
and
and if
This is usually denoted
f:\left(X,x0\right)\to\left(Y,y0\right).
Pointed spaces are important in
algebraic topology, particularly in
homotopy theory, where many constructions, such as the
fundamental group, depend on a choice of basepoint.
The pointed set concept is less important; it is anyway the case of a pointed discrete space.
Pointed spaces are often taken as a special case of the relative topology, where the subset is a single point. Thus, much of homotopy theory is usually developed on pointed spaces, and then moved to relative topologies in algebraic topology.
Category of pointed spaces
The class of all pointed spaces forms a category Top
with basepoint preserving continuous maps as
morphisms. Another way to think about this category is as the
comma category, (
Top) where
is any one point space and
Top is the
category of topological spaces. (This is also called a
coslice category denoted
Top.) Objects in this category are continuous maps
Such maps can be thought of as picking out a basepoint in
Morphisms in (
Top) are morphisms in
Top for which the following diagram
commutes:
It is easy to see that commutativity of the diagram is equivalent to the condition that
preserves basepoints.
As a pointed space,
is a
zero object in
Top
, while it is only a
terminal object in
Top.
There is a forgetful functor Top
Top which "forgets" which point is the basepoint. This functor has a
left adjoint which assigns to each topological space
the
disjoint union of
and a one-point space
whose single element is taken to be the basepoint.
Operations on pointed spaces
- A subspace of a pointed space
is a
topological subspace
which shares its basepoint with
so that the
inclusion map is basepoint preserving.
- One can form the quotient of a pointed space
under any
equivalence relation. The basepoint of the quotient is the image of the basepoint in
under the quotient map.
- One can form the product of two pointed spaces
as the
topological product
with
serving as the basepoint.
- The coproduct in the category of pointed spaces is the, which can be thought of as the 'one-point union' of spaces.
- The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such as compactly generated weak Hausdorff ones.
- The reduced suspension
of a pointed space
is (up to a
homeomorphism) the smash product of
and the pointed circle
- The reduced suspension is a functor from the category of pointed spaces to itself. This functor is left adjoint to the functor
taking a pointed space
to its
loop space
.
References