Let CX be the cone over a space
X
X
. Let a space
X
X
be first countable at
x
x
, then the following are equivalent: (1)
X
X
is locally simply connected at
x
x
; (2)
π
1
(
(
X
,
x
)
∨
(
X
,
x
)
,
x
)
{\pi _1}\left ( {\left ( {X,x} \right ) \vee \left ( {X,x} \right ),x} \right )
is naturally isomorphic to the free product
π
1
(
X
,
x
)
∗
π
1
(
X
,
x
)
{\pi _1}\left ( {X,x} \right ) * {\pi _1}\left ( {X,x} \right )
; (3)
π
1
(
(
C
X
,
x
)
∨
(
C
X
,
x
)
,
x
)
{\pi _1}\left ( {\left ( {CX,x} \right ) \vee \left ( {CX,x} \right ),x} \right )
is trivial. There exists a simply connected, locally simply connected Tychonoff space
X
X
with
x
∈
X
x \in X
, such that
(
X
,
x
)
∨
(
X
,
x
)
\left ( {X,x} \right ) \vee \left ( {X,x} \right )
is not simply connected.