In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a coherent topology is a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
that is uniquely determined by a family of
subspaces. Loosely speaking, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is coherent with a family of subspaces if it is a ''topological union'' of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
generated by a set of maps.
Definition
Let
be a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and let
be a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of subsets of
each having the subspace topology. (Typically
will be a
cover of
.) Then
is said to be coherent with
(or determined by
)
[ is also said to have the weak topology generated by This is a potentially confusing name since the adjectives and are used with opposite meanings by different authors. In modern usage the term is synonymous with ]initial topology
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
and is synonymous with final topology
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
. It is the final topology that is being discussed here. if the topology of
is recovered as the one coming from the
final topology
In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
coinduced by the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota ...
s
By definition, this is the
finest topology on (the underlying set of)
for which the inclusion maps are
continuous.
is coherent with
if either of the following two equivalent conditions holds:
* A subset
is
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
in
if and only if
is open in
for each
* A subset
is
closed in
if and only if
is closed in
for each
Given a topological space
and any family of subspaces
there is a unique topology on (the underlying set of)
that is coherent with
This topology will, in general, be
finer than the given topology on
Examples
* A topological space
is coherent with every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
of
More generally,
is coherent with any family of subsets whose interiors cover
As examples of this, a
weakly locally compact space is coherent with the family of its
compact subspaces. And a
locally connected space
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectednes ...
is coherent with the family of its connected subsets.
* A topological space
is coherent with every
locally finite closed cover of
* A
discrete space is coherent with every family of subspaces (including the
empty family).
* A topological space
is coherent with a
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of
if and only
is
homeomorphic to the
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of the elements of the partition.
*
Finitely generated spaces are those determined by the family of all
finite subspaces.
*
Compactly generated space
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition:
:A subsp ...
s are those determined by the family of all
compact subspaces.
* A
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
is coherent with its family of
-skeletons
Topological union
Let
be a family of (not necessarily
disjoint) topological spaces such that the
induced topologies agree on each
intersection
Assume further that
is closed in
for each
Then the topological union
is the
set-theoretic union
endowed with the final topology coinduced by the inclusion maps
. The inclusion maps will then be
topological embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
s and
will be coherent with the subspaces
Conversely, if
is a topological space and is coherent with a family of subspaces
that cover
then
is
homeomorphic to the topological union of the family
One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings.
One can also describe the topological union by means of the
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
. Specifically, if
is a topological union of the family
then
is homeomorphic to the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the disjoint union of the family
by the
equivalence relation
for all
; that is,
If the spaces
are all disjoint then the topological union is just the disjoint union.
Assume now that the set A is
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
, in a way compatible with inclusion:
whenever
. Then there is a unique map from
to
which is in fact a homeomorphism. Here
is the
direct (inductive) limit (
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
)
of
in the category
Top.
Properties
Let
be coherent with a family of subspaces
A function
from
to a topological space
is
continuous if and only if the restrictions
are continuous for each
This
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
characterizes coherent topologies in the sense that a space
is coherent with
if and only if this property holds for all spaces
and all functions
Let
be determined by a
cover Then
* If
is a
refinement of a cover
then
is determined by
In particular, if
is a
subcover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
of
is determined by
* If
is a refinement of
and each
is determined by the family of all
contained in
then
is determined by
* Let
be an open or closed
subspace of
or more generally a
locally closed subset of
Then
is determined by
* Let
be a
quotient map
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
. Then
is determined by
Let
be a
surjective map
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
and suppose
is determined by
For each
let
be the restriction of
to
Then
* If
is continuous and each
is a quotient map, then
is a quotient map.
*
is a
closed map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
(resp.
open map) if and only if each
is closed (resp. open).
Given a topological space
and a family of subspaces
there is a unique topology
on
that is coherent with
The topology
is
finer than the original topology
and
strictly finer if
was not coherent with
But the topologies
and
induce the same subspace topology on each of the
in the family
And the topology
is always coherent with
As an example of this last construction, if
is the collection of all compact subspaces of a topological space
the resulting topology
defines the
k-ification of
The spaces
and
have the same compact sets, with the same induced subspace topologies. And the k-ification
is compactly generated.
See also
*
Notes
References
*
* {{cite book, last=Willard, first=Stephen, title=General Topology, url=https://archive.org/details/generaltopology00will_0, url-access=registration, publisher=Addison-Wesley, location=Reading, Massachusetts, year=1970, isbn=0-486-43479-6, id=(Dover edition)
General topology