Hi to all!:)
I'm a new member, my name is Riccardo.
I've a problem with 1-manifolds, i have to prove a proposition but i can't do it.
Here is the extract of the book with notation and the proposition 1 that i have to prove:
DEFINITION. A 1-manifold is a second countable Hausdorff topological
space X such that X can be covered by open sets each of which is
homeomorphic either to the open interval (0,1) or the half-open
interval [0, 1). Sets of the first type will be called O-sets, of the second
type H-sets, of either type I-sets, and the corresponding homeomorphsms
to these intervals will be called O-charts,H-charts, and I-charts, respectively.
If X can be covered by O-sets it is a manifold without boundary, otherwise it
is a manifold with boundary.
From here on U and V will stand for I-sets in a 1-manifold and f and g will be
LEMMA. Suppose U∩V (set theoretic intersection) and U - V are
nonempty and let (x_n) be a sequence in U n V converging
to x in U - V (set theoretic difference). Then the sequence g(x_n)
has no limit point in g(V).
We say that U and V ouerlap if U∩V, U - V and V - U are nonempty.
DEFINITION. An open subinterval of (0,1) is lower if it is of the form (0, b) and
upper if it is of the form (a, 1). A subinterval which is either upper or lower is
called outer. It is easy to see that an open interval in (0,1) is outer if and only if it
contains a sequence with no limit point in (0,1). Similarly, in [0, 1), a subinterval is
called upper and outer if it is of the form (a, 1). (There are, by definition, no lower
open subintervals of [0, 1).) An open subinterval of [O,1) is outer if and only if it
contains a sequence with no limit point in [O,1).
PROPOSITION1. If U and V overlap and W is a component of U∩V, then f(W)
and g(W) are outer intervals.
Hint: Note that f(W) is a proper subinterval of f(U). Using the lemma show
that f(W) is an open interval. Then construct an appropriate sequence in f(W)
and use the lemma again.
Thank you in advance
Suppose has a limit point . Use the lemma on the sets and the sequence .