Connected space

In mathematical topology, there are different terms that describe the way the relationship of a topological space. Generally called a topological space X is connected if it is not possible for him in two disjoint, non-empty, split open subsets. A subspace of a topological space is called connected if it is connected with the induced topology.

A maximal connected subset of a topological space is called connected component.

• 4.1 Local contiguous
• 4.2 Local path-
• 4.3 Locally simply connected
• 4.4 Semi- local simply connected

Formal definition

For a topological space the following are equivalent:

A subset of a topological space is called connected if it is connected in the subspace topology; see in the example:

Example

Be. In words, is the disjoint union of two intervals. This amount is provided as usual with the induced topology ( subspace topology, trace topology). This means that the open reading amounts in the X, the amounts of the mold, one of which is in open set. A set is exactly then X is open if they can be written in an open set with X as a section.

The interval is open. So the intersection of with in X is open. This is just. So the set in X is open, although of course not in open.

Likewise, the interval is open. So the section of our space in X is open. This cut is now just the quantity. So is an open subset of the space.

Now we can write the space as a disjoint union of two open subsets in X, both of which are not empty. So is not contiguous.

This can alternatively be seen also as follows: The interval is completed. So is closed in X. This section is the amount that is completed in the X, although it is not completed.

Since, as explained above in X also is open, exists with a subset of which is at the same time both open and closed (in X ), but is not empty and not quite. So may not be contiguous.

Global context terms

The following terms always refer to the entire room, so are global properties:

Totally disconnected

A space is totally disconnected if it has no connected subset with more than one point, so if all connected components are one-pointed. Each discrete topological space is totally disconnected. In this case, the connected components are the ( one-point ) open. An example of a non-discrete totally disconnected space is the set of rational numbers with the induced topology.

Path-

A topological space is path (or path -connected or curve as connected), if for every pair of points from a path from to, ie a continuous mapping with and.

Path-connected spaces are always contiguous. Somewhat surprisingly, however, is at first glance, perhaps, that there are spaces that are connected but not path-connected. An example is the union of the graph of

A portion of the axis between -1 and 1, since is also a piece of the graph in any environment of zero, one can not be separated from the graph as an open subset of the axis; the amount is so connected. On the other hand, there is no path from a point on the graph to a point on the axis, so this association is not path-connected.

Simply connected

A space is simply connected if it is path and can be tightened every closed path to a point, ie is null homotopic. The second condition is equivalent to that the fundamental group is trivial.

Thus, in the adjacent figure, both the pink room C as well as his white complement " simply connected ", the former however only by the fact that a dividing line prevents the circumambulation of the white subscribed complement. In the lower part of image contrast, neither the orange room D still be white lined complement " simply connected " - you interpret D as a representation of the topology of a " sphere with four handles " that would complement the four "holes" of Henkel ball.

In contrast to subspaces of that, if they contain one or more non to the space belonging points ( "holes" ), thereby no longer " simply connected " are, this is true for subspaces of the first of all not: A space with the topology a ( all ) swiss cheese about still remains ( and independent of the number of holes in its interior ) " simply connected " because every closed path in such a space can be contracted to a point, bypassing the holes. If the room is on the other hand from a curve, such as a straight line completely crosses whose points are all not belong to the area will be the situation of Volltorus: A can no longer about the straight -closing path so be contracted to a single point.

N -connected

Is a non-negative integer, it means a topological space - connected if all homotopy groups are trivial. "0 -connected " is thus a synonym for " path connected ", and "1 -connected " means the same as " simply connected " in the sense defined above.

Contractible

A space X is contractible, if it is homotopy equivalent to a point, i.e. the identity of X is homotopic to a constant map. Contractible spaces have therefore from a topological point of view similar properties to a point, in particular, they are always simply connected. But the converse is not true: n - spheres with a fixed radius are not contractible, although they are contiguous for easy.

Local context terms

The following terms are local properties, so they make statements about the behavior in environments of points:

Locally connected

A space is locally connected if there is any neighborhood of a point a contiguous smaller neighborhood of this point. Each item then has a neighborhood base of connected sets.

A locally connected space may well consist of several connected components. But even a connected space need not be locally connected: The "comb" consisting of the union of the intervals, and is contiguous, but any sufficiently small neighborhood of the point contains infinitely many non-contiguous intervals.

Local path-

A space is locally path connected if every point has a local base of neighborhoods consisting of path-connected sets. A locally path-connected space is path-connected if and only if it is connected. The example given above with the graph of sin ( 1 / x) and the y-axis is therefore not local path connected. Adds also the x - axis should you get a coherent path-connected, locally path-connected but not space ( "Warsaw circle "). Furthermore, the " book" is path-connected, locally path-connected but not for all points on the perpendicular bisector with the exception of the intersection of all straight-line segments.

Locally simply connected

A space is locally simply connected if every neighborhood of a point contains a possibly small, simply-connected environment.

Manifolds are locally simply connected.

An example of a non- locally simply connected space are the " Hawaiian Earrings": The union of circles with radii as a subset of, so that all the circles touching at one point. Then each environment around the contact point contains a closed circuit, and therefore is not simply connected.

Semi local simply connected

A space X is semi- locally simply connected if every point has a neighborhood U such that each loop can pull together in U in X ( U they do not necessarily have to be contractible, therefore only semi- local).

Semi local simply connected a weaker condition than locally simply connected: A cone over the Hawaiian earrings is semi- locally simply connected, since each loop can be contracted through the cone tip. But he is ( for the same reason as the Hawaiian earrings itself) is not locally simply connected.