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Topological space

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In mathematics, a topological space is an ordered pair  where  is a set and  is a certain collection of subsets of  called the open sets or the topology of . The topology of  introduces an abstract structure of space in the set , which allows to define general notions such as of a point being surrounded by a set (by a neighborhood) or belonging to its boundary, of convergence of sequences of elements of , of connectedness, of a space or set being contractible, etc.

Definition

A topological space is an ordered pair  where  is a set and  is a collection of subsets of  (i.e., any element  is a subset of X) with the following three properties:

1.  and  (the empty set) are in 
2. The union of any family (infinite or otherwise) of elements of  is again in 
3. The intersection of two elements of  is again in 

Elements of the set  are called open sets of . We often simply write  instead of  once the topology  is established.

Once we have a topology in , we define the closed sets of  to be the complements (in ) of the open sets; the closed sets of  have the following characteristic properties:

1.  and  (the empty set) are closed
2. The intersection of any family of closed sets is closed
3. The union of two closed sets is closed

Alternatively, notice that we could have defined a structure of closed sets (having the properties above as axioms) and defined the open sets relative to that structure as complements of closed sets. Then such a family of open sets obeys the axioms for a topology; we obtain a one to one correspondence between topologies and structures of closed sets. Similarly, the axioms for systems of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology, and conversely --- every topology defines the systems of neighborhoods; for every set  we obtain a one to one correspondence between topologies in  and systems of neighborhoods in . These correspondences allow one to study the topological structure from different viewpoints.

The category of topological spaces

Given that topological spaces capture notions of geometry, a good notion of isomorphism in the category of topological spaces should require that equivalent spaces have equivalent topologies. The correct definition of morphisms in the category of topological spaces is the continuous homomorphism.

A function  is continuous if  is open in  for every open in . Continuity can be shown to be invariant with respect to the representation of the underlying topology; e.g., if  is closed in  for each closed subset  of Y, then  is continuous in the sense just defined, and conversely.

Isomorphisms in the category of topological spaces (often referred to as a homeomorphism) are bijective and continuous with continuous inverses.

The category of topological spaces has a number of nice properties; there is an initial object (the empty set), subobjects (the subspace topology) and quotient objects (the quotient topology), and products and coproducts exist as well. The necessary topologies to define on the latter two objects become clear immediately; if they're going to be universal in the category of topological spaces, then the topologies should be the coarsest making the canonical maps commute.

Examples

1. Let  where  denotes the set of real numbers. The open interval ]a, b[ (where a < b) is the set of all numbers between a and b:



Then a topology  can be defined on  to consist of  and all sets of the form:



where  is any arbitrary index set, and  and  are real numbers satisfying  for all . This is the familiar topology on  and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set  and in the next example another topology on , albeit a relatively obscure one, will be constructed.

2. Let  as before. Let  be a collection of subsets of  defined by the requirement that  if and only if  or  contains all except at most a finite number of real numbers. Then it is straightforward to verify that  defined in this way has the three properties required to be a topology on . This topology is known as the cofinite topology or Zariski topology.

3. Every metric  on  gives rise to a topology on . The open ball with centre  and radius  is defined to be the set



A set  is open if and only if for every , there is an open ball with centre  contained in . The resulting topology is called the topology induced by the metric . The standard topology on , discussed in Example 1, is induced by the metric .

4. For a given set , the family  is a topology: the indiscrete or weakest topology.

5. For a given set , the family  of all subsets of  is a topology: the discrete topology.

Neighborhoods

Given a topological space  of opens, we say that a subset  of  is a neighborhood of a point  if  contains an open set  containing the point  [1]

If  denotes the system of neighborhoods of  relative to the topology , then the following properties hold:

1.  is not empty for any 
2. If  is in  then 
3. The intersection of two elements of  is again in 
4. If  is in  and  contains , then  is again in 
5. If  is in  then there exists a  such that  is a subset of  and  for all 

Conversely, if we define a topology of neighborhoods on  via the above properties, then we can recover a topology of opens whose neighborhoods relative to that topology give rise to the neighborhood topology we started from:  is open if it is in  for all . Moreover, the opens relative to a topology of neighborhoods form a topology of opens whose neighborhoods are the same as those we started from. All this just means that a given topological space is the same, regardless of which axioms we choose to start from.

The neighborhood axioms lend themselves especially well to the study of topological abelian groups and topological rings because knowing the neighborhoods of any point is equivalent to knowing the neighborhoods of 0 (since the operations are presumed continuous). For example, the -adic topology on a ring  is Hausdorff if and only if , thus a topological property is equivalent to an algebraic property which becomes clear when thinking in terms of neighborhoods.

Bases and sub-bases

A basis for the topology  on X is a collection  of open sets such that every open set is a union of elements of . For example, in a metric space the open balls form a basis for the metric topology. A sub-basis  is a collection of open sets such that the finite intersections of elements of  form a basis for .

Some topological notions

This section introduces some important topological notions. Throughout,  will denote a topological space with the topology .

Partial list of topological notions
Closure

The closure in  of a subset  is the intersection of all closed sets of  which contain  as a subset.

Interior

The interior in  of a subset  is the union of all open sets of  which are contained in  as a subset.

Limit point

A point  is a limit point of a subset  of  if any open set in  containing  also contains a point  with . An equivalent definition is that  is a limit point of  if every neighbourhood of  contains a point  different from .

Open cover
A collection  of open sets of  is said to be an open cover for  if each point  belongs to at least one of the open sets in .
Path
A path  is a continuous function . The point  is said to be the starting point of  and  is said to be the end point. A path joins its starting point to its end point.
Hausdorff/separability property

 has the Hausdorff (or separability, or T2) property if for any pair  there exist disjoint sets  and  with  and .

Noetherianity

 is noetherian if it satisfies the descending chain condition for closed set: any descending chain of closed subsets  is eventually stationary; i.e., if there is an index  such that  for all .

Connectedness

 is connected if given any two disjoint open sets  and  such that , then either  or .

Path-connectedness
 is path-connected if for any pair  there exists a path joining  to . A path connected topological space is also connected, but the converse need not be true.
Compactness

 is said to be compact if any open cover of  has a finite sub-cover. That is, any open cover has a finite number of elements which again constitute an open cover for .

A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space.

Induced topologies

A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as induced topologies. Descriptions of some induced topologies are given below. Throughout,  will denote a topological space.

Some induced topologies

Relative topology

If  is a subset of  then open sets may be defined on  as sets of the form  where  is any open set in . The collection of all such open sets defines a topology on  called the relative topology of  as a subset of 

Quotient topology

If  is another set and  is a surjective function from  to  then open sets may be defined on  as subsets  of  such that . The collection of all such open sets defines a topology on  called the quotient topology induced by .

Product topology

If  is a family of topological spaces, then the product topology on the Cartesian product  has as sub-basis the sets of the form  where each  and  for all but finitely many .