With respect to the relative topology, is pathwise-connected.
Therefore it is connected, but it is not locally
pathwise-connected at any point of the open interval . Each disk centered at one of these points intersects
in a union of disjoint segments, which
form a disconnected set.

Let be the broom space formed by segments
of length
for all natural numbers ,
and place ,
, , ... one right after the other on the -axis. This will cover the half-open interval of the -axis (above figure). The space obtained by adding the point
(2,0) to this sequence of brooms is then connected
im kleinen at point (2,0), since each open neighborhood of (2,0) contains a closed disk whose radius is exactly formed by the basis
intervals of
for all sufficiently large .
Hence any two points contained in this disk are connected by a path composed of segments
of these broom spaces. On the other hand, point (2, 0) has no connected open neighborhood
since every open disk centered at (2,0) has no boundary and hence, unlike in the
case of a closed disk, it cannot end right at the vertex of some broom space. Therefore,
it must cut through some ,
which will be intersected in an union of disjoint segments, and these form a disconnected
set.