A *topological space* is a set *X*
together with a collection of subsets *OS*
the members of which are called *open*,
with the property that
(i) the union of an arbitrary collection of open sets is open, and
(ii) the intersection of a finite collection of open sets is open.
(In particular *X* is open, as is the empty set.)

**Examples**
(i) Metric spaces: there is a realvalued distance function
on *X × X* satisfying the triangle inequality;
an open set is a set containing balls with sufficiently small
radius around each of its points.

(ii) Ordered spaces: there is a total order on *X*;
an open set is a set containing for each point *y*
an entire orderinterval *(x,z)*, where *x < y < z*.

(iii) Discrete topology: every subset of *X* is open.

(iv) Indiscrete topology: the only open subsets of *X*
are the empty set and *X* itself.

A *map* or *continuous function*
from a topological space *(X,OX)* to a
topological space *(Y,OY)* is a function from
*X* to *Y* such that the preimage of
any member of *OY* is a member of *OX*.

A *homeomorphism* is a bijective map of which
the inverse is a map, too.

A *neighbourhood* of a set is an open set containing it.

A *closed* set is the complement of an open set.

A *basis* for the open sets is a collection of open sets
such that each open set is a union of some subcollection.
A *subbasis* for the open sets is a collection of open sets
such that one obtains a basis by taking finite intersections.
In order to check that a given function is continuous, it suffices
to check that the inverse images of the members of a subbasis
for the open sets are open again.

The *interior* of a set *A* is the largest open
set contained in *A*. The *closure* of *A*
is the smallest closed set containing *A*. The *boundary*
of *A* is the intersection of its closure and the closure
of its complement.

**Example**
Let *(X,OX)* be the closed unit interval [-1,1]
with usual topology and let *(Y,OY)* be the space
obtained from it by identifying the points 0 and 1.
Let *p* be the quotient map. Then the restriction *p'*
of *p* to the half-open interval [-1,1) is a bijection
and is continuous, but its inverse is not continuous,
so *p'* is not a homeomorphism.

**Exercise**
(i) Let *A* be a subset of a topological space *X*.
How many different subsets can I make starting from *A* and
repeatedly using the closure and interior operations?
Answer.

(ii) Let *A* be a subset of a topological space *X*.
How many different subsets can I make starting from *A* and
repeatedly using the boundary and union operations?
Answer.

Let *A* be a set, and *F* a collection of functions
from *A* to topological spaces. The topology on *A*
defined by *F* is the weakest topology (i.e., the smallest
collection *OA*) for which all these functions become
continuous.

Similarly, let *B* be a set, and *F* a collection
of functions from topological spaces to *B*.
The topology on *B* defined by *F* is the strongest
topology (i.e., the largest collection *OB*) for which all
these functions become continuous.

Given a topological space *(X,OX)* and a
subset *A*, we may (and will) consider it a topological space
in its own right (a *subspace*) by giving it the topology
defined by the inclusion map.

Given an arbitrary collection of topological spaces *(Xi,OXi)*,
their (Cartesian) product is the topological space with as point set
the Cartesian product of the sets *Xi*, and topology defined
by the projection maps.

Given a topological space *(X,OX)* and a function *f*
from *X* to a set *B*, we call the topology on *B*
determined by *f* the *quotient topology*, and *f*
the corresponding *quotient map*.
Frequently *B* will be the set of equivalence classes in
*X* of some equivalence relation *R*. In this case
the quotient space is denoted *X/R*.

A topological space is called *Hausdorff* (or (T2))
when any two points have disjoint neighbourhoods.
(A weaker requirement called (T1) is that every singleton
is a closed subset.)
A topological space is called *normal*
when any two disjoint closed sets have
disjoint neighbourhoods.
A topological space is called *metric* when there is a
*distance function* determining the topology
(i.e., open balls for the metric are open sets, and conversely,
if a point *x* lies in an open set *U* then for
some positive *e* the ball with radius *e* around
*x* is contained in *U*.

A metric space is normal since *{x|d(x,A) < d(x,B)}*
and *{x|d(x,A) > d(x,B)}* are disjoint neighbourhoods
of the disjoint closed sets *A* and *B*.

A Hausdorff space *X* is normal if and only if for each
pair of disjoint closed sets *A* and *B* there
exists a map *f* from *X* to the unit interval *I*
that is identically 0 on *A* and identically 1 on *B*.
(Urysohn)

A topological space is called *compact*
when every open cover (i.e., covering with open sets)
has a finite subcover.

A compact subset of a Hausdorff space is closed.

A closed subset of a compact space is compact.

The continuous image of a compact space is compact again.

A topological space is called *paracompact*
when every open cover has a locally finite refinement.

A *filterbase* is a collection of nonempty sets such that
the intersection of any two contains a third.
We say that a filterbase converges to a point if each
neighbourhood of the point contains an element of the filterbase.
We say that a filterbase accumulates at a point if each
neighbourhood of the point meets an element of the filterbase.

A space is Hausdorff iff each convergent filterbase converges to exactly one point.

A space is compact iff each filterbase has an accumulation point.

A space is compact iff each maximal filterbase converges.

**Exercise**
Prove: the cartesian product of compact spaces is compact.
(Tychonoff)

A topological space is called *connected*
when the only subsets that are both open and closed
are the empty set and the entire space.
A *(connected) component* of a topological space
is a maximal connected subset.

The continuous image of a connected space is connected again.

In particular, an image of the closed unit interval [0,1]
(sometimes called an *arc* or a *path*) is connected.
A topological space is called *path-connected*
or *arcwise connected* when any two of its points
can be joined by an arc.
(Note: below we shall use the word path for the mapping from [0,1]
into some space, rather than for its image.)

The intersection of all open-and-closed subsets containing a
given point *x* is called the *quasi-component* of *x*.
The partition of a topological space into quasi-components is coarser
than the partition into components.

**Exercise**
Prove: The closure of a connected set in an arbitrary topological space
is again connected. Components and quasicomponents of a topological space
are closed.

**Exercise**
Prove: Let *X* be connected, and *A* a connected subset,
and *Q* either open-and-closed, or a component, or a quasicomponent
in *X\A*. Then *X\Q* is connected.
Answer

**Exercise**
(i) Construct a topological space that is connected but not path-connected.
(ii) Construct a topological space with a quasicomponent that is not connected.
(iii) Show that in a compact space components and quasicomponents coincide.

**Warning** There exist strange objects, like connected
Hausdorff spaces that become totally disconnected (all components
are singletons) when one removes a well-chosen point, or
connected metric spaces without nontrivial open connected subspaces.
Also, there need not be any non-constant maps from [0,1] to a
connected topological space (indeed, there exist countable
connected Hausdorff spaces), and in such a case all path-components
are singletons.

**Problem** Show that there exists a connected subset of the
Euclidean plane such that all its path-components are singletons.

A topological space is said to be *locally P* for some
property *P* when for each point *x* and
neighbourhood *U* of *x* there is a set *A*
contained in *U* and containing a neighbourhood of *x*
that has property *P*.

For example, the Euclidean plane is locally compact but not compact.

**Exercise**
(i) Give an example of a space that is connected but not locally connected.
(ii) Show that if *X* is connected and *A* a compact
nonempty proper subspace, then each component of *A*
meets the boundary of *A*.

Let *C(X,Y)* be the set of all maps from *X* into *Y*.
This set has several natural topologies.
First of all, there is the product topology, where this is regarded
as a subspace of the product of *|X|* copies of *Y*.
Here a basic open set constrains the values at finitely many points
to an open set. Much finer is the *compact-open* topology
(also called *c*-topology), where a subbasic open set
contains the functions mapping a given compact subset of *X*
into a given open subset of *Y*. Below we shall assume that
*C(X,Y)* has the compact-open topology.

Let *Y* be locally compact and *X,Z* Hausdorff.
Then the composition map from *C(X,Y) × C(Y,Z)* to
*C(X,Z)* is continuous. In particular, under these assumptions,
the evaluation map from *C(Y,Z) × Y* to *Z*
is continuous.

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