MAT 530 Topology

Fall 2004

__Instructor:__ Vladlen
Timorin

e-mail: mailto:timorin@math.sunysb.edu

Office: 4-102 Phone: 2-8265

Office hours: WF 9-11 or by appointment

__Textbook:__ Munkres *Topology*, 2nd edition, Prentice Hall 1999

__Syllabus __(from the Graduate Student Handbook):

Basic point set
topology

- Metric Spaces
- Topological spaces and continuous maps
- Comparison of topologies
- Separation axioms and limits
- Countability axioms, the Urysohn metrization theorem
- Compactness and paracompactness, the Tychonoff theorem
- Connectedness
- Product spaces
- Function spaces and their topologies, Ascoli's theorem

- Fundamental group
- Fundamental group of
*S*^{n}; examples of fundamental groups of surfaces - Seifert-van Kampen theorem
- Classification of covering spaces, universal covering spaces; examples
- Homotopy; essential and inessential maps

First I tried to illustrate the following folklore definition of topology: topology studies the properties of geometric objects that remain unchanged under all continuous deformations. We gave examples of several topological invariants (without any proofs). I made some remarks on the topological nature of the famous Euler's formula in combinatorics of convex polyhedra.

The next were the formal definitions of metric and topological spaces, bases and subbases in topological spaces (i.e., a description of different ways to define topology on a set). Of course, the most interesting examples of topological spaces are metric spaces. Examples of metric spaces: Euclidean spaces, max-norm on a finite-dimensional real vector space, p-adic metrics on the field of rational numbers, etc.

Some other examples of topological spaces: the 3 essentially different topologies on a 2-point set, the order topology of a linearly ordered set, we also know how to define topology on a partially ordered set such that any pair of elements admits a lower bound.

*Closed sets* are complemets to the open sets; they form the so called
closed topology and provide an alternative way to define topological spaces. We
defined the closure and the interior of any subset in a topological space, and
studied some properties of the closure.

The notion of *continuous maps* is probably the most important in all
topology. It generalizes continuous functions of real arguments. We gave a
topological definition of continuity that does not appeal to a metric structure
or any other additional structures and thus can be applied to any topological
spaces. *Homeomorphisms* are bijective continuous maps with continuous
inverses. Intuitively, the continuity prevents "cutting", and the continuity of
the inverse map prevents "glueing". Homeomorphic topological spaces have the
same topological properties.

A *quotient space*, i.e. the set of equivalence classes of some
equivalence relation on a topological space, can be equipped with a natural
topology - *the quotient topology*. This is the largest (finest, strongest)
topology such that the canonical projection (from the space to the
quotient-space) is continuous. The direct (Cartesian) product of two topological
spaces also carries a natural *direct product topology* that is the
smallest (coarsest, weakest) topology such that the projections to the factors
are continuous.

A topological space is called *connected* if it does not split into a
disjoint union of two open (hence closed) subsets. E.g. the real line and all
intervals are connected, but the union of 2 disjoint open intervals is always
disconnected. All connected subsets of the real line are open intervals (that
may be empty and may be infinite) with, possibly, some of the ends attached. Any
topological space splits into *connected components* that are maximal
connected subsets. This gives a description of all open subsets of the real
line: these are countable (or finite) disjoint unions of open intervals (or
rays). The image of a connected space under a continuous map is also connected.
This is a generalization of the Intermediate Value Theorem from Calculus.
Connected components are always closed, but not always open. A counterexample is
the set of all rational numbers with the topology induced from the reals (which
is the same as the order topology) --- all rationals are separate connected
components, but they are not open. A space all of whose connected components are
open, is called *locally connected*. To be locally connected, it is enough
to have at least one connected neighborhood for each point.

A *path* connecting two points of a topological space is a continuous
map from a segment to this space such that the ends of the segment get mapped to
given two points. If every pair of points of a topological space can be
connected by a path, then the space is said to be *path connected*. Any
path connected space is connected. Any space splits to path components, but they
are not necessarily closed. There are examples of connected spaces that are not
path connected. E.g. the graph of sin(1/*x*) together with the
*y*-axis. A space all of whose path components are open (in other words,
any point has a path connected neighborhood) is called *locally path
connected*. A connected locally path connected space is path connected.

A topological space is said to be *compact* if any open covering of it
contains a finite subcovering. A subset of a finite dimensional Euclidean space
is compact if and only if it is closed and bounded. Closed subsets of compact
spaces are compact in the subspace topology, the product of two compact spaces
is also compact. For a __metric space__ *X* the compactness is
equivalent to any of the following statements:

*X*satisfies the*Bolzano-Weierstass property*, i.e. any sequence contains a convergent subsequence.*X*is*complete*(any Cauchy sequence converges) and*totally bounded*(for any positive epsilon there is a finite epsilon-net).

A continuous function on a compact metric space is uniformly continuos.

The image of a compact space under a continuous map is compact. It follows that a continuous function on a compact space always attains its maximum and minimum.

The space *C(X,Y)* of all continuous functions of a topological space
*X* to a metric space *Y* carries the natural *uniform* metric.
Convergence with respect to this metric is exactly the uniform convergence. If
*Y* is complete, then the space *C(X,Y)* is also complete. It follows
that if a sequence of continuous functions converges uniformly to some function,
then the limit is continuous.

A compact subset of a metric space is always closed and bounded. But in
infinite dimensional spaces, being closed and bounded is not enough for being
compact. E.g. the unit ball in *C([0,1], R)* is not compact.

*A*is closed,*A*is bounded,*A*is*equicontinuous*.

*The Baire theorem* states that in a complete metric space, the
intersection of countably many open dense sets is dense ("dense" means that the
closure is the whole space). We say that "almost all points" of a complete
metric space (in the sense of the Baire category) satisfy a certain property, if
the set of points with this property is the intersection of countably many open
dense sets. For example,

- "almost all" real numbers are irrational,
- the point-wise limit of a sequence of continuous functions is "almost everywhere" continuous,
- "almost every" continuous function is nowhere differentiable,
- for "almost any continuous function", its values at all rational points are irrational.

Some set theoretic preparation for the proof of the Tychonoff theorem is here [pdf, ps]. This is about the axiom of choice, the Zorn lemma and the Zermelo theorem.

A *filter* on a set *X* is a collection of subsets of *X* such
that

- it does not contain the empty set
- it is closed under finite intersections
- together with any its set
*A*, it contains any bigger set (i.e., any set containing the set*A*).

The existence of a nonprincipal unltrafilter allows to define the field of
*hyperreal numbers* that are used in the *non-standard analysis*. Let
us fix a nonprincipal ultrafilter on the set of natural numbers. Define a
hyperreal number to be a sequence of real numbers. Two such sequences are said
to be equal if they are equal element-wise on some "big" set (a set from our
ultrafilter). In a sense, all properties of real numbers have exact analogs for
hyperreals. We can find the difference only while coparing the hyperreals with
the reals (that are represented as constant sequences). And it turns out that
there are infinitely small hyperreals (infinitesimals) that are positive and
smaller than any positive real number. Infinitesimals allow to define
derivatives, integrals, etc. without using the limits.

Let us say that a collection of sets has the *finite intersection
property* if the intersection of any its finite subcollection is nonempty.
The compactness of a to pological space is equivalent to any the following
statements:

- If a collection of closed sets has a finite intersection property, then its intersection is nonempty.
- For any filter, the intersection of closures of all its elements is nonempty.
- For any ultrafilter, the intersection of closures of all its elements is nonempty.

Consider the Cartesian product *X* of infinitely many topological spaces
*X _{a}*. The

Two subsets of a topological space can be *separated by open
neighborhoods* if they have disjoint open neighborhoods. *Separation
axioms* for topological spaces state that certain pairs of subsets can be
separated by open neighborhoods. A space is called

*Hausdorff*if any pair of different points can be separated,*regular*if any point can be separated from any closed subset not containing this point, and all one-pont sets are closed*normal*if any two disjoint closed subsets can be separated, and all one-point sets are closed.

The __Urysohn lemma__ states that for a normal topological space *X*
and two disjoint closed subsets *A* and *B* of it, there exists a
continuous function from *X* to [0,1] that is 0 on *A* and 1 on
*B*. This condition guarantees that normal spaces have enough continuous
functions. Suppose that a topological space *X* is normal and has a
countable basis. Then there exist a countable family of continuous functions on
*X* taking values in the unit segment [0,1] and satisfying the following
property: for any point of X and any open neighborhood of this point, there is a
function from our family such that this function is positive at the given point
and is identically zero outside of the given neighborhood. In other words, a
countably family of continuous functions is enough to separate any point from
any closed set not containing this point. This follows from the Urysohn lemma
and from the existence of a countable basis. A countable family of continuous
functions with the property discussed above, gives a metric on *X* and an
embedding of *X* into l_{2}. This fact is known as the __Urysohn
metrization theorem__.

To prove that some topological space is metrizable, we need to make sure that
the topology on it is not too large. The countability axioms serve that purpose.
__The first countability axiom__ states that every point has a countable
basis of neighborhoods, i.e. a collection of neighborhoods such that for any
neighborhood our point there is a subneighborhood from our collection. __The
second countability axiom__ states that there is a countable basis. Every
metric space satisfies the first countability axiom, but there are metric spaces
that do not admit a countable basis (e.g. real numbers with the discrete
metric). The first countability axiom allows to use limits of sequences. For
general topological spaces, the limit of every convergent sequence from some set
belongs to the closure of this set. But the converse is not true. There can be
some points in the closure that can not be reached by any countable sequence (so
that one needs to take uncountable sequences, say, indexed by a large
well-ordered set, to reach those points). An example is given by any
well-ordered set (with respect to the order topology) containing uncountable
initial segments. Any regular second countable set is normal. Hence in the
Urysohn theorem, the normality assumption can be replaced with just regularity
assumption.

A collection of sets is called *locally finite* if any point has a
neighborhood intersecting only finitely many sets from this collection. A
*countably locally finite* collection is a countable union of locally
finite collections. The __Nagata-Smirnov metrization theorem__ (which we did
not prove in class) gives necessary and sufficient conditions on a topological
space to be metrizable. These conditions are formulated in purely topological
terms. Namely, a topological space is metrizable if and only if it is regular
and has a countably locally finite basis.

One collection of sets is said to be a *refinement* of another
collection, if every set from the first collection is contained in some set from
the second collection. A Hausdorff topological space is *paracompact*, if
any open covering of it admits a locally finite refinement that covers the
space. Clearly, any Hausdorff compact space is paracompact. By a very nontrivial
theorem due to Stone, every metric space is also paracompact. Paracompact spaces
are normal, in particular, regular. Any open covering of any paracompact space
admits a *partition of unity*, i.e. a family of continuous functions that
take values in [0,1], sum up to 1 and have small enough supports (the support of
any function from this family must lie in some open set from the given open
covering). Partitions of unity are very useful in geometry and topology, e.g.
they help to prove embedding theorems. As an example of an embedding theorem, we
proved that every compact topological manifold can be embedded into some affine
space (a topological space with a countable basis is said to be a topological
*n*-manifold if every point of this space has a neighborhood homeomorphic
to **R**^{n}).

We started with examples of good topological spaces. These include: Euclidean
spaces, spheres, real projective spaces, complex projective spaces, etc. Next we
defined *homotopy* of paths (or a continuous deformation). Two paths are
*homotopic* if there exists a homotopy between them. The relation of being
homotopic is an equivalence relation. The equivalence classes with respect to
this relation are called *homotopy classes*. A loop in a topological space
is a path that begins and ends at the same point (this point is called the
*base point*). Let us fix a base point. The homotopy classes of loops with
this base point form so called the *fundamental group* of the topological
space. The group structure is given by "multiplication of two loops" (the
product of two loops is the loop formed by first going along one loop, and then
going along the other loop). A continuous map from one path connected space to
another path connected space is called a *covering* if it looks locally,
over a small neighborhood, like the standard projection of the direct product of
this neighborhood with a discrete space onto this neighborhood. Examples of
coverings:

- The real lines covers the circle, with infinitely many preimages of each point. This covering is used to prove that the fundamental group of the circle is freely generated by a single element (i.e., it is isomorphic to the additive group of all integers)
- A
*k*-fold covering of the circle by the circle exists for every positive integer*k*. The 2-fold covering of this type can be visualized as the projection from the boundary of the Moebius band to its middle circle. - There is a 2-fold covering of a projective space by a sphere of the same dimension.

One of the earliest and the most important topological theorems is Euler's formula.

The definition of topological spaces is due to Felix Hausdorff. Algebraic topology was founded by Henry Poincare. His works were motivated by problems from classical mechanics, as well as by earlier insights of Bernhard Riemann.

Some other topologists whose names appeared in this course: Encyclopaedia articles: