§ 4. Differential Manifolds
245
1 3'
2'
−1'
1
0
−1
1
2
4'
3
4
Fig. 13.8.
is everywhere dense in T
2 , the union Δ of the segments obtained in K is
everywhere dense in K. The map F being, however, a homeomorphism, if
it is restricted to a neighbourhood of 0 in D, then the set of points of the
trajectory Z = F (Δ) in the neighbourhood of the point (1, 1) of T
2 is seen
to be homeomorphic to the intersection of Δ and of a neighbourhood of 0
in K; this intersection is composed of infinitely many pairwise disjoint line
segments. In the neighbourhood of the origin in T
2 , the trajectory Z can,
therefore, be decomposed into an infinitely countable number of excellent
pairwise disjoint arcs of curve. Hence if Z is equipped with the topology of T
2 ,
we get a space, which while being connected, is not locally connected,
63 nor
even locally compact since the intersection of Δ with a closed neighbourhood
of 0 in K is obviously not closed and even less compact. All is now proved.
The non-closed geodesics of the torus are examples of immersions f :
X −→ Y which, while being injective, are not homeomorphisms from X onto
63 A topological space Z is said to be locally connected if, for every x ∈ Z and
every neighbourhood V of x in Z, there is a connected neighbourhood U of a
such that U ⊂ V : existence of arbitrarily small connected neighbourhoods.
245
1 3'
2'
−1'
1
0
−1
1
2
4'
3
4
Fig. 13.8.
is everywhere dense in T
2 , the union Δ of the segments obtained in K is
everywhere dense in K. The map F being, however, a homeomorphism, if
it is restricted to a neighbourhood of 0 in D, then the set of points of the
trajectory Z = F (Δ) in the neighbourhood of the point (1, 1) of T
2 is seen
to be homeomorphic to the intersection of Δ and of a neighbourhood of 0
in K; this intersection is composed of infinitely many pairwise disjoint line
segments. In the neighbourhood of the origin in T
2 , the trajectory Z can,
therefore, be decomposed into an infinitely countable number of excellent
pairwise disjoint arcs of curve. Hence if Z is equipped with the topology of T
2 ,
we get a space, which while being connected, is not locally connected,
63 nor
even locally compact since the intersection of Δ with a closed neighbourhood
of 0 in K is obviously not closed and even less compact. All is now proved.
The non-closed geodesics of the torus are examples of immersions f :
X −→ Y which, while being injective, are not homeomorphisms from X onto
63 A topological space Z is said to be locally connected if, for every x ∈ Z and
every neighbourhood V of x in Z, there is a connected neighbourhood U of a
such that U ⊂ V : existence of arbitrarily small connected neighbourhoods.