Contiguous space

Connected and non-connected subspaces of ℝ²: A is simply connected, B (the entire blue) and the complements of A and B are not.

In mathematical topology , there are several terms that the way the relationship of a topological space describe. In general, a topological space is said to be connected if it is not possible to divide it into two disjoint , non-empty, open subsets. A subspace of a topological space is called connected if it is connected under the induced topology . ${\displaystyle X}$

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

Formal definition

The following statements are equivalent for a topological space : ${\displaystyle {\big (}X,{\mathcal {O}}{\big )}}$

1. ${\displaystyle X}$ is contiguous.
2. ${\displaystyle X}$cannot be decomposed into two disjoint, non-empty open sets :

   ${\displaystyle \forall O_{1},O_{2}\in {\mathcal {O}},O_{1}\neq \emptyset ,O_{2}\neq \emptyset :O_{1}\cap O_{2}=\emptyset \Rightarrow O_{1}\cup O_{2}\neq X}$

3. ${\displaystyle X}$cannot be decomposed into two disjoint non-empty closed sets :

   ${\displaystyle \forall O_{1},O_{2}\in {\mathcal {O}},O_{1}\neq \emptyset ,O_{2}\neq \emptyset :(X\setminus O_{1})\cap (X\setminus O_{2})=\emptyset \Rightarrow (X\setminus O_{1})\cup (X\setminus O_{2})\neq X}$

4. ${\displaystyle X}$and are the only two sets that are open and closed at the same time. ${\displaystyle \emptyset }$
5. The only blank- bordered sets are and . ${\displaystyle X}$ ${\displaystyle \emptyset }$
6. ${\displaystyle X}$cannot be written as the union of two non-empty separate sets .
7. Every continuous mapping of into a discrete topological space is constant. ${\displaystyle X}$
8. Every locally constant function of in is constant. ${\displaystyle X}$

A subset of a topological space is called contiguous if it is a contiguous space in the subspace topology (see following example).

Some authors do not consider the empty topological space to be connected (although it fulfills the eight equivalent conditions). This has certain advantages, for example a space is connected with this definition if and only if it has exactly one connected component.

example

Be . So in words is the disjoint union of two intervals. This amount is, as usual with that of induced topology ( subspace topology , track topology provided). This means that those in open sets are just those sets of the form , where one is in open set . A lot is so if and in open when they are in section of a in open set with can be written. ${\displaystyle X:=\left[0,1\right[\cup \left[3,4\right[\subseteq \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle V\cap X}$ ${\displaystyle V}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$

The interval is in open. So the cut from with in is open . This just results . So the crowd is in , although of course it's not in . ${\displaystyle V_{1}:=\left]-1,2\right[}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V_{1}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle [0,1[}$ ${\displaystyle [0,1[}$ ${\displaystyle X}$ ${\displaystyle [0,1[}$ ${\displaystyle \mathbb {R} }$

The interval in is also open. So the intersection of our space in is open. This cut is now the crowd . So is an open subset of space . ${\displaystyle V_{2}:=\left]2,5\right[}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V_{2}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle [3,4[}$ ${\displaystyle [3,4[}$ ${\displaystyle X}$

With this one can write the space as a disjoint union of two in open subsets, both of which are not empty. So is not connected. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

Alternatively, this can also be seen as follows: The interval is completed in. So in complete. This cut is the amount , so in complete, although not complete. ${\displaystyle [0,2]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle [0,2]\cap X}$ ${\displaystyle X}$ ${\displaystyle [0,1[}$ ${\displaystyle [0,1[}$ ${\displaystyle X}$ ${\displaystyle [0,1[}$ ${\displaystyle \mathbb {R} }$

Since, as explained above, in is also open, there is a subset of that is both open and closed (in ) at the same time , but is not empty and also not whole . So it can not be connected. ${\displaystyle [0,1[}$ ${\displaystyle X}$ ${\displaystyle [0,1[}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

Connected component

In a topological space, the connected component of a point is equal to the union of all those connected subspaces which contain this point, i.e. the largest of all connected subspaces to which this point belongs .

particularities

Special features of connected subspaces of the real coordinate space

In the real coordinate space , connected subspaces have several peculiarities. Two of them deserve special mention.

Connected subspaces of real numbers

These are the real intervals . The following applies:

The connected subspaces of are the real intervals of each type . In detail, it is about the empty set , the one-point subsets and all open, half-open, closed, restricted and unrestricted intervals with at least two points, including themselves. ${\displaystyle \mathbb {R} }$ ${\displaystyle \emptyset }$ ${\displaystyle \mathbb {R} }$

Indeed, it can show that a subspace     and only then is connected if for any two points also apply. ${\displaystyle T\subseteq \mathbb {R} }$ ${\displaystyle a\;,\;b\in T}$ ${\displaystyle [a,b]\subseteq T}$

Areas

With regard to the connected sub-areas of the , the following particularity is particularly noteworthy: ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$

A non-empty open set forms a connected subspace (and thus a region ) if and only if it is path-connected (see below).

It is even more strictly true that in such an area every two points can always be connected by a route that lies entirely in this area .

Special feature of compact metric spaces

This peculiarity consists in the following property:

If a metric space is compact , then it is connected if and only if two of its points for each are -chained in the sense that finitely many points exist with and as well . ${\displaystyle (X,d)}$ ${\displaystyle a,b\in X}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \epsilon }$ ${\displaystyle x_{1},\ldots ,x_{n}\in X\;(n=n(a,b,\epsilon )\in \mathbb {N} )}$ ${\displaystyle a=x_{1}}$ ${\displaystyle b=x_{n}}$ ${\displaystyle d(x_{i},x_{i+1})<\epsilon \;(i=1,\ldots ,n-1)}$

Global contexts

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

Totally incoherent

→ Main article : Totally incoherent space

A space is totally disconnected if it has no connected subset with more than one point, i.e. if all connected components are one-point. Any discrete topological space is totally disconnected. In this case the (one-point) connected components are open. An example of a non-discrete, totally disconnected space is the set of rational numbers with the topology induced by. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$

Path-related

This subspace of R ² is path because two of its points are connected by a path.

This subspace of R ² is indeed connected, but not path-connected.

A topological space is path-connected (or path-connected or curve as contiguous or pathwise connected ), if for each pair of points , from a way of to is, d. H. a continuous mapping with and . ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle p\colon [0,1]\to X}$ ${\displaystyle p(0)=x}$ ${\displaystyle p(1)=y}$

Spaces that are connected to each other are always connected. At first glance, however, it may be a bit surprising that there are rooms that are contiguous, but not contiguous. An example is the union of the graph of

${\displaystyle (0,\infty )\to \mathbb {R} ,\quad x\mapsto \sin(1/x)}$

with a section of the -axis between −1 and 1, with the topology induced by. Since there is also a piece of the graph in every neighborhood of zero, the -axis cannot be separated from the graph as an open subset; the set is therefore connected. On the other hand, there is no path from a point on the graph to a point on the -axis, so this union is not path-connected. ${\displaystyle y}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle y}$ ${\displaystyle y}$

A maximum path-connected subset of a topological space is called a path-connection component .

Simply connected

Connected and non-connected subspaces of R²  : C and its complement (without the black line) are simply connected; D and its complement, however, are not.

A space is simply connected if it is path-connected and every closed path can be contracted to a point, i.e. H. is null homotop . The second condition is equivalent to the fact that the fundamental group is trivial.

In the illustration opposite, both the pink-colored space and its white complement are “simply connected”, the former only because a dividing line prevents the complement drawn in white from going around. In the lower part of the picture, on the other hand, neither the orange-colored space nor its white-drawn complement are “simply connected” - if one interprets the topology of a “ball with four handles” as a representation, the complement would be the four “holes” of the handle ball. ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle D}$

In contrast to subspaces of the , which, as soon as they contain one or more points ("holes") that do not belong to the space, are no longer "simply connected", this does not apply to subspaces of the : A space with the topology A (whole) Swiss cheese, for example, remains “simply connected” (regardless of the number of holes in its interior) because every closed path in such a space can be drawn together to a point by bypassing the holes. On the other hand, if the room is surrounded by a curve, B. a straight line, completely crossed , the points of which do not all belong to the space, the situation of the full torus arises : A path that closes around the straight line can no longer be reduced to a single point. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$

n -contiguous

Is a non-negative integer, it means a topological space -connected if all homotopy groups for are trivial. “0-connected” is therefore a synonym for “path-connected”, and “1-connected” means the same as “simply connected” in the sense defined above. ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle \pi _{\,k\,}(X)}$ ${\displaystyle 0\leq k\leq n}$

Contractible

→ Main article : Contractible space

A space X is contractible if it is homotopy equivalent to a point, i.e. the identity on X is homotopy to a constant map. From a topological point of view, contractible spaces therefore have similar properties to a point, in particular they are always simply connected . But the converse is not true: n-spheres of fixed radius are not contractible even though they are simply connected. ${\displaystyle n\geq 2}$

Local contextual terms

Comb: connected but not locally connected

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

Locally contiguous

→ Main article : Locally connected

A space is locally connected if there is a connected smaller neighborhood of this point for every neighborhood of a point. Each point then has a neighborhood base made up of connected sets.

A locally connected space can consist of several connected components. But also a connected space does not necessarily have to be locally connected: The “comb”, consisting of the union of the intervals , and , is connected, but every sufficiently small neighborhood of the point contains an infinite number of non-connected intervals. ${\displaystyle \{0\}\times [0,1]}$ ${\displaystyle [0,1]\times \{0\}}$ ${\displaystyle [0,1]\times \{1/n\}}$ ${\displaystyle (1,0)}$

Book: route-related, but not locally connected

Locally connected to the road

A space is locally connected in a path or locally connected in an arc if each point has a neighborhood base which consists of path-connected neighborhoods. A locally path-connected space is path-connected if and only if it is connected. The example given above with the graph of and the -axis is therefore not locally path- related . If you also add the -axis you get a contiguous, path-connected, but not locally path-connected space (" Warsaw Circle "). Furthermore, the “book” is path-connected, but not locally connected for all points on the perpendicular with the exception of the intersection of all straight lines. ${\displaystyle \sin(1/x)}$ ${\displaystyle y}$ ${\displaystyle x}$

Hawaiian earrings: not semi-locally simply connected and not locally simply connected either

Locally simply connected

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

Manifolds are locally simply connected.

The Hawaiian earrings are an example of a space that is not simply connected locally : The union of circles with radii as a subset of the so that all circles touch at one point. Then every environment around the point of contact contains a closed circle and is therefore not simply connected. ${\displaystyle 1/n}$ ${\displaystyle \mathbb {R} ^{2}}$

Semilocal simply connected

A space is semi- locally simply connected if every point has a neighborhood so that every loop in in can be contracted (in it does not necessarily have to be contractible in, therefore only semi- locally). ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle U}$

Semilocal simply connected is a weaker condition than locally simply connected : A cone above the Hawaiian earrings is semilocal simply connected, since each loop can be drawn together over the tip of the cone. But it is not locally simply connected (for the same reason as the Hawaiian earrings themselves).

literature

• P. Alexandroff , H. Hopf : Topologie (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 45 ). First volume. Corrected reprint. Springer Verlag, Berlin / Heidelberg / New York 1974 ( MR0185557 ).
• Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set-theoretical and algebraic topology (=  Berlin study series on mathematics . Volume 15 ). Heldermann Verlag, Lemgo 2006, ISBN 3-88538-115-X ( MR2172813 ).
• Lutz Führer : General topology with applications . Vieweg Verlag, Braunschweig 1977, ISBN 3-528-03059-3 .
• Klaus Jänich : Topology (=  Springer textbook ). 7th edition. Springer Verlag, Berlin (among others) 2001, ISBN 3-540-41284-0 .
• James R. Munkres : Topology . 2nd Edition. Prentice Hall, Upper Saddle River NJ 2000, ISBN 0-13-181629-2 .
• B. v. Querenburg : Set theoretical topology . 2nd, revised and expanded edition. Springer Verlag, Berlin / Heidelberg / New York 1979 ( MR0639901 ).
• Willi Rinow : Textbook of Topology (=  university books for mathematics . Volume 79 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1975 ( MR0514884 ).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).

Web links

• Eric W. Weisstein : Connected Set . In: MathWorld (English).
• Connected Set Entry in the Springer Encyclopedia of Mathematics

Individual evidence

1. ↑ This is a consequence of the chain theorem .
2. ↑ Thorsten Camps et al .: Introduction to set theoretical and algebraic topology. 2006, p. 94
3. ↑ P. Alexandroff, H. Hopf: Topology. 1974, p. 49
4. ^ Lutz Führer: General topology with applications. 1977, p. 86
5. ↑ Horst Schubert: Topology. 1975, p. 39
6. ↑ Camps et al., Op.cit., P. 88
7. ↑ Schubert, op.cit., P. 38
8. ↑ P. Alexandroff, H. Hopf: Topology. 1974, p. 50
9. ^ Willi Rinow: Textbook of Topology. 1975, p. 150
10. ↑ Camps et al., Op.cit., P. 98
11. ↑ Führer, op.cit., P. 125
12. ↑ B. v. Querenburg: Set theoretical topology. 1979, p. 96