Comb Space
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In mathematics, particularly
topology 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 ho ...
, a comb space is a particular subspace of \R^2 that resembles a
comb A comb is a tool consisting of a shaft that holds a row of teeth for pulling through the hair to clean, untangle, or style it. Combs have been used since Prehistory, prehistoric times, having been discovered in very refined forms from settlemen ...
. 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 ...
. 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 \R^2 with its
standard 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 vecto ...
and let ''K'' be the set \. The set ''C'' defined by: :(\ \times ,1) \cup (K \times ,1 \cup ( ,1\times \) considered as a subspace of \R^2 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: :\ \cup (K \times ,1 \cup ( ,1\times \) . This is the comb space with the line segment \ \times
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 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 path connected, or locally connected. 2. The deleted comb space, D, is connected: ::Let E be the comb space without \ \times (0,1">Locally connected space">locally path connected, or locally connected. 2. The deleted comb space, D, is connected: ::Let E be the comb space without \ \times (0,1. E is also path connected and the closure of E is the comb space. As E \subset D \subset 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 ...
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'' (Y ...
and closed 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; American and British English spelling differences, see spelling differences) is a geographically localised community ...
''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 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, ...
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'') × \R and ''B'' = (''r'', +∞) × \R will form a
separation Separation may refer to: Films * ''Separation'' (1967 film), a British feature film written by and starring Jane Arden and directed by Jack Bond * ''La Séparation'', 1994 French film * ''A Separation'', 2011 Iranian film * ''Separation'' (20 ...
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 defor ...
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 deform ...
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 In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point. For any cardinal number \kappa, the \kappa-hedgehog space is formed by taking the disjoint union of \kappa real unit intervals identified at ...
*
Infinite broom In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also r ...
*
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, connected ...
*
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, ...
*
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)