Topological area

a topological area is the fundamental the subject of the partial discipline topology of mathematics. Itconsists of any quantity, to which by specification of a topology in such a way specified an abstract mathematical regional structure is impressed.

Definition

a topology is a family <math> \ mathfrak {T}< /math> of subsets of the fundamental set X, which is called open, so that the following axioms are fulfilled:

• The empty quantity and the fundamental set X are open quantities.
• The average finally many open quantitiesis an open quantity.
• The combination as many as desired open quantities is an open quantity.

A quantity of X as well as a topology on X is called topological area. A subset of X, whose complement is an open quantity, is called final.

A topology <math> \ mathfrak {T} _1< /math> than a topology math \ <mathfrak> {T is finer} _2< /math>, if each open quantity of <math> \ mathfrak {T} _2< /math> also openly in <math> \ mathfrak {T} _1< /math> is. <math> \ mathfrak {T} _2< /math> than math \ mathfrak <{>T} _1 /math is called then< rougher>.

Further terms in connection with topological areas are summarized in the topology glossary.

Environments

a substantial term in topological areas is that the environment. A subset of U of a topological area <math> (X, \ mathfrak {T})< /math> environment of the point x from X is called, if an open quantity <math> O< /math> with <math> x \ in O \ U /math< subset> existed. The system thatEnvironments of x /math often becomes <with> math \ mathfrak {U} (x<)> designated. To the environments the following characteristics apply:

1. Is <math> U< /math> in <math> \ mathfrak {U} (x)< /math> and <math> U \ subset U'\ subset X< /math>, then is also <math> U'< /math> in <math> \ mathfrak {U}< /math>.
2. Math <U_j> is \ in \ mathfrak {U} (x)< /math> to <math> 1 \ leq j \ leq n< /math>, then applies also <math> \ bigcap_ {for j=1} ^ {n} U_j \ in \ mathfrak {U} (x)< /math>.
3. Is <math> U \ in \ mathfrak {U} (x)< /math>, then applies <math> for x \ in U< /math>.
4. To everyone <math> U< /math> in <math> \ mathfrak {U} (x)< /math> exists <math> a V< /math> in <math> \ mathfrak {U} (x)< /math>, so that <math> U \ in \ mathfrak {U} (y)< /math> for everyone <math> y< /math> in <math> V< /math> applies.

<math> \ mathfrak U (x)< /math> a filter is and therefore also environment filter is called.

One arranges against it each point x a quantity of X a quantity system <math> \ mathfrak {U} (x)< /math> too, so that above conditions are fulfilled, then there is a clearly determined topology on X, so that for each x the system <math> \ mathfrak {U} (x)< /math> the environment system from x is. A quantity is inthis case exactly openly if it contains an environment of this point with each of its points also. (This sentence explains the use of the word openly for the mathematical term defined above.)

examples

1. on each fundamental set exist astrivial examples of topologies:
   1. The indiscrete (or chaotic or becoming lumpy) topology, which contains only the empty quantity and the fundamental set.
   2. The discrete topology, which contains all subsets.
2. The system of the open subsets of a metric area is a topology.
3. Assomewhat more unusual example exists to M /math on <an infinite> quantity< math> (e.g. the quantity <math> \ mathbb {N}< /math> the natural numbers) the kofinite topology: The empty quantity as well as each subset of math M </math>,< whose> complement contains only finally many elements, are open.

manner of speaking

Regarding geometrical applications the elements of the fundamental set are often called points.

production of topological areas

• of each subset of Y of the fundamental set X of an existing topology can be assigned a Unterraumtopologie. Are the open quantities straightthe cuts of the open quantities of the existing topology with the subset of Y.
• With each family of topological areas can be assigned to the product of the fundamental sets the product topology. With finite products the open quantities become by the products of the open quantitiesformed.
• A quotient topology develops through „for sticking together “from topological areas.

literature

• Klaus Jänich: Topology. 6. Edition, Springer, Berlin 1999, ISBN 3540653619
• Boto v. Traverse castle: Set-theoretical topology. 3. Edition. Springer, Berlin 2001, ISBN 3540677909
• refuge thrust ore: Topology.Teubner, Stuttgartto 1964, ISBN 3519122006

see also

• area (mathematics)