|
|
| Line 1: |
Line 1: |
| In [[general topology]] and related areas of [[mathematics]], the '''initial topology''' (or '''weak topology''' or '''limit topology''' or '''inductive topology''') on a [[Set (mathematics)|set]] <math>X</math>, with respect to a family of functions on <math>X</math>, is the [[coarsest topology]] on ''X'' which makes those functions [[continuous function (topology)|continuous]].
| | Hello! Allow me to start by saying my name - Vernita although i don't like when people use my full identity. For years she's been working like a financial expert. The thing I enjoy most crochet and now I have enough time to take on new important things. Georgia is his birth place. He's not godd at design but you may want to look his website: http://www.partybus.com/philadelphia/pa/united-states |
| | |
| The [[subspace topology]] and [[product topology]] constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.
| |
| | |
| The [[duality (mathematics)|dual]] construction is called the [[final topology]].
| |
| | |
| ==Definition==
| |
| | |
| Given a set ''X'' and an [[indexed family]] (''Y''<sub>''i''</sub>)<sub>''i''∈''I''</sub> of [[topological space]]s with functions
| |
| :<math>f_i: X \to Y_i</math>
| |
| the initial topology τ on <math>X</math> is the [[coarsest topology]] on ''X'' such that each
| |
| :<math>f_i: (X,\tau) \to Y_i</math>
| |
| is [[continuous function (topology)|continuous]].
| |
| | |
| Explicitly, the initial topology may be described as the topology [[subbase|generated by]] sets of the form <math>f_i^{-1}(U)</math>, where <math>U</math> is an [[open set]] in <math>Y_i</math>. The sets <math>f_i^{-1}(U)</math> are often called [[cylinder set]]s.
| |
| If ''I'' contains just one element, all the open sets of <math>(X,\tau)</math> are cylinder sets.
| |
| | |
| ==Examples==
| |
| | |
| Several topological constructions can be regarded as special cases of the initial topology.
| |
| * The [[subspace topology]] is the initial topology on the subspace with respect to the [[inclusion map]].
| |
| * The [[product topology]] is the initial topology with respect to the family of [[projection map]]s.
| |
| * The [[inverse limit]] of any [[inverse system]] of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms.
| |
| * The [[weak topology]] on a [[locally convex space]] is the initial topology with respect to the [[continuous linear form]]s of its [[dual space]].
| |
| * Given a [[indexed family|family]] of topologies {τ<sub>''i''</sub>} on a fixed set ''X'' the initial topology on ''X'' with respect to the functions id<sub>''X''</sub> : ''X'' → (''X'', τ<sub>''i''</sub>) is the [[supremum]] (or join) of the topologies {τ<sub>''i''</sub>} in the [[lattice of topologies]] on ''X''. That is, the initial topology τ is the topology generated by the [[union (set theory)|union]] of the topologies {τ<sub>''i''</sub>}.
| |
| * A topological space is [[completely regular]] if and only if it has the initial topology with respect to its family of ([[bounded function|bounded]]) real-valued continuous functions.
| |
| * Every topological space ''X'' has the initial topology with respect to the family of continuous functions from ''X'' to the [[Sierpiński space]].
| |
| | |
| ==Properties==
| |
| ===Characteristic property===
| |
| | |
| The initial topology on ''X'' can be characterized by the following [[universal property]]: a function <math>g</math> from some space <math>Z</math> to <math>X</math> is continuous if and only if <math>f_i \circ g</math> is continuous for each ''i'' ∈ ''I''. | |
| [[Image:InitialTopology-01.png|center|Characteristic property of the initial topology]]
| |
| | |
| ===Evaluation===
| |
| | |
| By the universal property of the [[product topology]] we know that any family of continuous maps ''f''<sub>''i''</sub> : ''X'' → ''Y''<sub>''i''</sub> determines a unique continuous map
| |
| :<math>f\colon X \to \prod_i Y_i\,</math>
| |
| This map is known as the '''evaluation map'''.
| |
| | |
| A family of maps {''f''<sub>''i''</sub>: ''X'' → ''Y''<sub>''i''</sub>} is said to ''[[Separating set|separate points]]'' in ''X'' if for all ''x'' ≠ ''y'' in ''X'' there exists some ''i'' such that ''f''<sub>''i''</sub>(''x'') ≠ ''f''<sub>''i''</sub>(''y''). Clearly, the family {''f''<sub>''i''</sub>} separates points if and only if the associated evaluation map ''f'' is [[injective]].
| |
| | |
| The evaluation map ''f'' will be a [[topological embedding]] if and only if ''X'' has the initial topology determined by the maps {''f''<sub>''i''</sub>} and this family of maps separates points in ''X''.
| |
| | |
| ===Separating points from closed sets===
| |
| | |
| If a space ''X'' comes equipped with a topology, it is often useful to know whether or not the topology on ''X'' is the initial topology induced by some family of maps on ''X''. This section gives a sufficient (but not necessary) condition.
| |
| | |
| A family of maps {''f''<sub>''i''</sub>: ''X'' → ''Y''<sub>''i''</sub>} ''separates points from closed sets'' in ''X'' if for all [[closed set]]s ''A'' in ''X'' and all ''x'' not in ''A'', there exists some ''i'' such that
| |
| :<math>f_i(x)\notin \operatorname{cl}(f_i(A))</math>
| |
| where ''cl'' denoting the [[closure (topology)|closure operator]].
| |
| | |
| :'''Theorem'''. A family of continuous maps {''f''<sub>''i''</sub>: ''X'' → ''Y''<sub>''i''</sub>} separates points from closed sets if and only if the cylinder sets <math>f_i^{-1}(U)</math>, for ''U'' open in ''Y''<sub>i</sub>, form a [[base (topology)|base for the topology]] on ''X''. | |
| | |
| It follows that whenever {''f''<sub>''i''</sub>} separates points from closed sets, the space ''X'' has the initial topology induced by the maps {''f''<sub>''i''</sub>}. The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.
| |
| | |
| If the space ''X'' is a [[T0 space|T<sub>0</sub> space]], then any collection of maps {''f''<sub>i</sub>} which separate points from closed sets in ''X'' must also separate points. In this case, the evaluation map will be an embedding.
| |
| | |
| == Categorical description ==
| |
| | |
| In the language of [[category theory]], the initial topology construction can be described as follows. Let ''Y'' be the [[functor]] from a [[discrete category]] ''J'' to the [[category of topological spaces]] '''Top''' which selects the spaces ''Y''<sub>''j''</sub> for ''j'' in ''J''. Let ''U'' be the usual [[forgetful functor]] from '''Top''' to '''Set'''. The maps {''f''<sub>''j''</sub>} can then be thought of as a [[cone (category theory)|cone]] from ''X'' to ''UY''. That is, (''X'', ''f'') is an object of Cone(''UY'')—the [[category of cones]] to ''UY''.
| |
| | |
| The characteristic property of the initial topology is equivalent to the statement that there exists a [[universal morphism]] from the forgetful functor
| |
| :''U''′ : Cone(''Y'') → Cone(''UY'')
| |
| to the cone (''X'', ''f''). By placing the initial topology on ''X'' we therefore obtain a functor
| |
| :''I'' : Cone(''UY'') → Cone(''Y'')
| |
| which is [[adjoint functor|right adjoint]] to the forgetful functor ''U''′. In fact, ''I'' is a right-inverse to ''U''′ since ''U''′''I'' is the identity functor on Cone(''UY'').
| |
| | |
| == See also ==
| |
| * [[Final topology]]
| |
| | |
| ==References==
| |
| | |
| *{{cite book | last = Willard | first = Stephen | title = General Topology | publisher = Addison-Wesley | location = Reading, Massachusetts | year = 1970 | isbn = 0-486-43479-6 (Dover edition)}}
| |
| *{{planetmath reference|id=7368|title=Initial topology}}
| |
| *{{planetmath reference|id=7504|title=Product topology and subspace topology}}
| |
| | |
| [[Category:General topology]]
| |
Hello! Allow me to start by saying my name - Vernita although i don't like when people use my full identity. For years she's been working like a financial expert. The thing I enjoy most crochet and now I have enough time to take on new important things. Georgia is his birth place. He's not godd at design but you may want to look his website: http://www.partybus.com/philadelphia/pa/united-states