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In mathematics, particularly 
.
1._The_comb_space,_C,_is_path_connected_and_Contractible_space">contractibletopology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
, a comb space is a particular subspace of that resembles a comb. The comb space has properties that serve as a number of counterexamples
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
. The topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the functi ...
has similar properties to the comb space. The deleted comb space is a variation on the comb space.
Formal definition
Consider with itsstandard topology
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
and let ''K'' be the set . The set ''C'' defined by:
:
considered as a subspace of equipped with the subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
is known as the comb space. The deleted comb space, D, is defined by:
:.
This is the comb space with the line segment Topological properties
The comb space and the deleted comb space have some interesting topological properties mostly related to the notion of locally connected space">connectedness In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be .... 1. The comb space, C, is path connected and Contractible space">contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
, but not locally contractible, Locally connected space">locally path connected, or locally connected. 2. The deleted comb space, D, is connected: ::Let E be the comb space without . E is also path connected and the Closure (topology), closure of E is the comb space. As E D the closure of E, where E is connected, the deleted comb space is also connected. 3. The deleted comb space is not path connected since there is no
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
from (0,1) to (0,0):
::Suppose there is a path from ''p'' = (0, 1) to the point (0, 0) in ''D''. Let ''ƒ'' : , 1nbsp;→ ''D'' be this path. We shall prove that ''ƒ'' −1 is both open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
and closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
in , 1contradicting the connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be ...
of this set. Clearly we have ''ƒ'' −1 is closed in , 1by the continuity of ''ƒ''. To prove that ''ƒ'' −1 is open, we proceed as follows: Choose a neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area ...
''V'' (open in R2) about ''p'' that doesn’t intersect the ''x''–axis. Suppose ''x'' is an arbitrary point in ''ƒ'' −1. Clearly, ''f''(''x'') = ''p''. Then since ''f'' −1(''V'') is open, there is a basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
element ''U'' containing ''x'' such that ''ƒ''(''U'') is a subset of ''V''. We assert that ''ƒ''(''U'') = which will mean that ''U'' is an open subset of ''ƒ'' −1 containing ''x''. Since ''x'' was arbitrary, ''ƒ'' −1 will then be open. We know that ''U'' is connected since it is a basis element for the order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, t ...
on , 1 Therefore, ''ƒ''(''U'') is connected. Suppose ''ƒ''(''U'') contains a point ''s'' other than ''p''. Then ''s'' = (1/''n'', ''z'') must belong to ''D''. Choose ''r'' such that 1/(''n'' + 1) < ''r'' < 1/''n''. Since ''ƒ''(''U'') does not intersect the ''x''-axis, the sets ''A'' = (−∞, ''r'') × and ''B'' = (''r'', +∞) × will form a separation on ''f''(''U''); contradicting the connectedness of ''f''(''U''). Therefore, ''f'' −1 is both open and closed in , 1 This is a contradiction.
4. The comb space is homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
to a point but does not admit a deformation retract
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...
onto a point for every choice of basepoint.
See also
*Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
* Hedgehog space
* Infinite broom
* List of topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, ...
* Locally connected space
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness ...
* Order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, t ...
* Topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the functi ...
References
* *{{Cite journal, title=Connectedness, series=Encyclopedic Dictionary of Mathematics, publisher=Mathematical Society of Japan, editor=Kiyosi Itô Topological spaces Trees (topology)