• Source: Uniformly disconnected space

In mathematics, a uniformly disconnected space is a metric space



(
X
,
d
)


{\displaystyle (X,d)}

for which there exists



λ
>
0


{\displaystyle \lambda >0}


such that no pair of distinct points



x
,
y
∈
X


{\displaystyle x,y\in X}

can be connected by a



λ


{\displaystyle \lambda }

-chain.
A



λ


{\displaystyle \lambda }

-chain between



x


{\displaystyle x}

and



y


{\displaystyle y}

is a sequence of points




x
=

x

0


,

x

1


,
…
,

x

n


=
y


{\displaystyle x=x_{0},x_{1},\ldots ,x_{n}=y}

in



X


{\displaystyle X}

such that



d
(

x

i


,

x

i
+
1


)
≤
λ
d
(
x
,
y
)
,
∀
i
∈
{
0
,
…
,
n
}


{\displaystyle d(x_{i},x_{i+1})\leq \lambda d(x,y),\forall i\in \{0,\ldots ,n\}}

.




Properties


Uniform disconnectedness is invariant under quasi-Möbius maps.




References

Kata Kunci Pencarian:

• Uniformly disconnected space
• Uniformly connected space
• Uniform property
• Connected space
• Discrete space
• Compact space
• T1 space
• Glossary of general topology
• Trivial topology
• Universe