Suppose has a limit point . Use the lemma on the sets and the sequence .
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:
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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
associated I-charts.
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.
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Thank you in advance