A topology space is a set with
a collection of open sets that satisfy the following
properties: 1. and . 2. if , then . 3.
if , then
.
A topological space is denoted as .
Basis and subbasis for a
Topology
A basis for a
topology on is a collection of
sets such that: 1. For each , there exists a basis element such that . 2. If and , then there exists a basis element such that .
A topology generated by a basis is the collection of all
unions of elements of .
Formally, .
A subbasis for a topology on
is a collection of sets
such that the collection of finite intersections of elements of forms a basis for a topology
on . Formally, a subbasis satisfies: 1. For each , there exists a subbasis element
such that . 2. The collection of finite
intersections of elements of generates a basis for the
topology on .
A topology generated by a subbasis is denoted as .
Countability,
Compactness and Separation Axioms
A topological space is first countable if every
point has a countable local base, meaning for each point , there exists a countable
collection of open sets such that for any
open set containing , there exists some such that .
A topological space is second countable if it has a
countable base, meaning there exists a countable collection of open sets
such that every open
set in the topology can be expressed as a union of sets from .
A topological space is compact if every open cover
has a finite subcover, meaning for any collection of open sets such that , there exists a
finite subset such
that .
A topological space is
(Kolmogorov) if for any two distinct points , there exists an open set
containing one of the points but not the other. Intuitively, this means
that a pair of point can be one-sidely distinguished by open sets for
one of them.
A topological space is
(Frechet) if for any two distinct points , there exist open sets and such that and , and and . Intuitively, this means
that points can be one-sidely separated by open sets for both of
them.
A topological space is
(Hausdorff) if for any two distinct points , there exist disjoint open
sets and such that and . This means that points can be
separated by disjoint open sets.
A topological space is regular if it is and for every point and closed set not containing , there exist disjoint open sets and such that and .
This means that points can be separated from closed sets by disjoint
open sets.
A topological space is normal if for any two
disjoint closed sets and , there exist disjoint open sets and such that and . This means that closed sets can be separated by disjoint
open sets.
A topological space is
(regular Hausdorff) if it is both regular and Hausdorff. This means that
points can be separated from closed sets by disjoint open sets, and
distinct points can be separated by disjoint open sets. spaces are also normal.
Important properties of these separation axioms include: - . - A single point set is closed in a space. - A space is normal if it is also regular
since a point is closed in a
space.
Functions and Homeomorphisms
A function between
two topological spaces and is open if for every open set
, the image
is open in . In other words,
A function is
closed if for every closed set , the image is closed in . In other words,
A function between topological spaces is
continuous if for every open set , the preimage is open in . In other words,
A function is a homeomorphism if it is a
continuous bijection and its inverse is also continuous.
In other words, both and are continuous functions.
A homeomorphism establishes a topological equivalence between the two
spaces, meaning they have the same topological properties.
Connectedness and Path
Connectedness
A topological space is connected if it cannot be
expressed as the union of two disjoint non-empty open sets. In other
words, there are no two open sets
and such that , , and both and are non-empty.
A topological space is path connected if for any two
points , there exists a
continuous function (path) such that and
. This means that there is
a continuous path connecting any two points in the space.
A topological space is locally path connected if
every point has a neighborhood base consisting of path connected sets.
This means that for each point , there exists a collection of open sets such that for each
, is path connected and contains , and for any open set containing , there exists some such that .
A topological space is locally connected if every
point has a neighborhood base consisting of connected sets. This means
that for each point , there
exists a collection of open sets such that for each
, is connected and contains , and for any open set containing , there exists some such that .
Common Topologies
Fineness and coarseness of topologies are important concepts in
topology. A topology
is finer than another topology on the same set if every open set in is also an open set in
. In other words,
. Conversely, is coarser than if is finer than .
The discrete topology on a set is the finest topology, where every
subset of is open. Formally,
, the power set of .
The indiscrete topology (or trivial topology) on a
set is the coarsest topology,
where only the empty set and the entire set are open. Formally, .
The quotient topology on a set with an equivalence relation is defined as follows: 1. The
quotient set is , where is the
equivalence class of . 2. A subset
is open in the
quotient topology if and only if its preimage under the natural
projection map
is open in . That is, is open if is open in .
In other words, the quotient topology is the coarsest topology on
such that the natural
projection map is
continuous.
The product topology on a product of topological
spaces is defined
as follows: 1. The product space is . 2. A subset is open in the product topology if it can be
expressed as a union of sets of the form , where is open in for each , and for all but finitely many . In other words, the product topology
is generated by the basis consisting of all products of open sets, where
only finitely many factors are not the entire space.
The box topology on a product of topological spaces
is defined
similarly to the product topology, but without the restriction that only
finitely many factors can be different from the entire space: 1. The box
space is .
2. A subset is open
in the box topology if it can be expressed as a union of sets of the
form , where
is open in for each . In other words, the box topology
is generated by the basis consisting of all products of open sets, where
each factor can be any open set in the corresponding space.
The subspace topology on a subset of a topological space
is defined as
follows: 1. The subspace is . 2. A
subset is open in the
subspace topology if it can be expressed as for some open set . In other words, is open in the subspace topology if it
is the intersection of with an
open set in .