Metric and Topological spaces JOC MT4522 2003/4

Summary

Metric spaces

A metric on a set X is a map d: X X R such that:
d(x, y) 0 with d(x, y) = 0 if and only if x = y; d(x, y) = d(y, x),
(The triangle inequality) d(x, y) + d(y, z) d(x, z).

Some examples of metric spaces are:
R with its usual metric d(x, y) = |x - y|
R², R³, ... with the usual (Pythagorean) metric d₂ or a variety of other metrics like d₁ (the taxi-cab metric) or d (the supremum metric).
Spaces of functions like C[0, 1] with metrics like d₁ , d₂ , d .
The discrete metric on any set: d(x, y) = 1 if x y.

An (open) -neighbourhood of a point p in a metric space X is
V(p) = {x X | d(x, p) < }.
A subset A of a metric space X is called open if every point p A has some -neighbourhood lying completely inside A.
A union of arbitrarily many open sets is open. An intersection of finitely many open sets is open.

A sequence (a_n) in a metric space converges to a limit if
(a) given > 0, there exists N such that n > N d(a_n, ) < ,
or (b) every -neighbourhood of p contains all but finitely many terms of the sequence,
or (c) every open set containing p contains all but finitely many terms of the sequence.

A point is a limit point of a subset A of a metric space if
(a) is the limit of a sequence in A which is not ultimately constant,
or (b) every -neighbourhood of meets A in a point ,
or (c) every open set containing meets A in a point .

A subset which contains all its limit points is called closed.
A subset A of a metric space X is closed if and only if X - A is open.

A function f : X Y between metric spaces is continuous at p X if
(a) given > 0 there exists > 0 such that d(x, p) < d(f (x), f (p)) < ,
or (b) every e-neighbourhood of f (p) contains the image of some -neighbourhood of p,
or (c) the inverse image of every open set of Y containing f (p) is an open set of X.

A function which is continuous at p maps sequences which converge to p into sequences which converge to f (p).
"Continuous functions preserve convergence."

Topological spaces

A topological space is a set X together with a set of subsets called "open sets" such that:
the subsets and X and is closed under arbitrary unions and finite intersections.

Closed sets are complements of open sets.
A basis for a topology is a set of subsets such that any set in can be written as a union of sets in . In a metric space, the -neighbourhoods form a basis for the topology.

Some examples of topological spaces are:
Any metric space with the open sets defined as above,
The trivial topology on any set X: = {, X },
Certain topologies on finite sets. e.g. the Sierpinski topology:
X = {a, b}, = {, {a}, {a, b}},
The cofinite (or Zariski) topology in which proper closed sets are the finite sets,
The co-countable topology in which proper closed sets are the countable sets.

The interior int(A) of a set A in a topological space is the largest open subset of A.
The closure cl(A) of a subset A is the smallest closed subset containing A.
A function f : X Y between topological spaces is continuous if f ^-1(A) is open in X whenever A is open in Y.
A continuous bijection whose inverse function is also continuous is called a homeomorphism or topological isomorphism.

Various topologies

If A is a subset of a topological space X, the subspace topology on A is the topology whose open subsets are all of the form A U for U open in X.
If X and Y are topological spaces the product topology on X Y has as basis the products of open sets in X with open sets in Y.
If X is a topological space and ~ is an equivalence relation on X then the identification topology on the set X/~ of equivalence classes is the topology in which the open sets of X/~ are the images of open sets of X under the natural map p: X X/~.

The subspace topology is the weakest topology (fewest open sets) on the subset in which the inclusion map of the subset is continuous.
The product topology is the weakest topology on the product in which the projection maps from X Y to X and to Y are both continuous.
The identification topology is the strongest topology (most open sets) on X/~ in which the natural map p is continuous.

Separation axioms

A topological space is called Hausdorff if every pair of points can be "separated" by open sets. That is, given x y one can find disjoint open sets U and V with x U and y V.
A topological space is called normal if every pair of disjoint closed sets can be "separated" by open sets. That is, given closed sets A and B with A B = , one can find disjoint open sets U and V with A U and B V.
Every metric space is both Hausdorff and normal.
In Hausdorff spaces sequences have at most one limit.

Connectedness

A topological space X is connected if
(a) one cannot write it as a union of disjoint open subsets,
or (b) the only sets of the topology which are both open and closed are X and .

The continuous image of a connected space is connected.
If A and B are connected (in the subspace topology) and A B then A B is connected.
The only connected subsets of R (with its usual topology) are intervals.
From this one can deduce the Intermediate Value Theorem.
Maximal connected subsets of a topological space are called its components.
A path from p to q in a topological space X is a continuous map from the unit interval [0, 1] to X with (0) = p and (1) = q. A space X is called pathwise-connected if every pair of points in X can be connected by a path.
A pathwise-connected space is connected, but not necessarily vice-versa.

Compactness

A topological space X is called compact if every open covering of X can be reduced to a finite sub-covering.
That is, if X = with A_i open, then X = A_i1 A_i2 ... A_in for some i₁ , i₂ , ... i_n I.

The interval [0, 1] in R (with its usual topology) is compact.
The continuous image of a compact space is compact.
Form this one may deduce that a continuous real-valued function on a closed bounded interval is bounded and attains its bounds.
A closed subset of a compact space is compact (in the subspace topology).
(Heine-Borel theorem) Any closed bounded subset of R (with its usual metric) is compact.
A compact subset of a Hausdorff space is closed.
A compact Hausdorff space is normal.
(Tychonoff's theorem) A product of compact spaces is compact.

A metric space is sequentially compact if every sequence contains a convergent sub-sequence.
A metric space is sequentially compact if (and only if) it is compact.
(Fom this one can deduce the Bolzano-Weierstrass theorem)