Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.

Definition and equivalent formulations

A topological space

X

{\displaystyle X}

is called simply connected if it is path-connected and any loop in

X

{\displaystyle X}

defined by

f

:

S

1

→

X

{\displaystyle f:S^{1}\to X}

can be contracted to a point: there exists a continuous map

F

:

D

2

→

X

{\displaystyle F:D^{2}\to X}

such that

F

{\displaystyle F}

restricted to

S

1

{\displaystyle S^{1}}

is

f

.

{\displaystyle f.}

Here,

S

1

{\displaystyle S^{1}}

and

D

2

{\displaystyle D^{2}}

denotes the unit circle and closed unit disk in the Euclidean plane respectively.

An equivalent formulation is this:

X

{\displaystyle X}

is simply connected if and only if it is path-connected, and whenever

p

:

[

0

,

1

]

→

X

{\displaystyle p:[0,1]\to X}

and

q

:

[

0

,

1

]

→

X

{\displaystyle q:[0,1]\to X}

are two paths (that is, continuous maps) with the same start and endpoint (

p

(

0

)

=

q

(

0

)

{\displaystyle p(0)=q(0)}

and

p

(

1

)

=

q

(

1

)

{\displaystyle p(1)=q(1)}

), then

p

{\displaystyle p}

can be continuously deformed into

q

{\displaystyle q}

while keeping both endpoints fixed. Explicitly, there exists a homotopy

F

:

[

0

,

1

]

×

[

0

,

1

]

→

X

{\displaystyle F:[0,1]\times [0,1]\to X}

such that

F

(

x

,

0

)

=

p

(

x

)

{\displaystyle F(x,0)=p(x)}

and

F

(

x

,

1

)

=

q

(

x

)

.

{\displaystyle F(x,1)=q(x).}

A topological space

X

{\displaystyle X}

is simply connected if and only if

X

{\displaystyle X}

is path-connected and the fundamental group of

X

{\displaystyle X}

at each point is trivial, i.e. consists only of the identity element. Similarly,

X

{\displaystyle X}

is simply connected if and only if for all points

x

,

y

∈

X

,

{\displaystyle x,y\in X,}

the set of morphisms

Hom

Π

(

X

)

⁡

(

x

,

y

)

{\displaystyle \operatorname {Hom} _{\Pi (X)}(x,y)}

in the fundamental groupoid of

X

{\displaystyle X}

has only one element.