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In mathematics, a dendrite is a certain type of 
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
that may be characterized either as a locally connected
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 ...
dendroid or equivalently as a locally connected continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
that contains no simple closed curves.
Importance
Dendrites may be used to model certain types ofJulia set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
. For example, if 0 is pre-periodic, but not periodic, under the function , then the Julia set of is a dendrite: connected, without interior..
References
See also
*Misiurewicz point
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point (mathematics), critical point is strictly p ...
*Real tree In mathematics, real trees (also called \mathbb R-trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the s ...
, a related concept defined using metric spaces instead of topological spaces
*Dendroid (topology)
In mathematics, a dendroid is a type of topological space, satisfying the properties that it is hereditarily unicoherent (meaning that every subcontinuum of ''X'' is unicoherent), arcwise connected, and forms a continuum. The term dendroid was ...
and unicoherent space In mathematics, a unicoherent space is a topological space X that is connected and in which the following property holds:
For any closed, connected A, B \subset X with X=A \cup B, the intersection A \cap B is connected.
For example, any closed i ...
, two more general types of tree-like topological space
Continuum theory
Trees (topology)
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