**Metric and Topological spaces JOC MT4522 2003/4**

**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*|

**R**^{2}, **R**^{3}, ... with the usual (Pythagorean) metric *d*_{2} or a variety of other metrics like *d*_{1} (the taxi-cab metric) or *d*_{} (the supremum metric).

Spaces of functions like *C*[0, 1] with metrics like *d*_{1} , *d*_{2} , *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 *i*_{1} , *i*_{2} , ... *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)