Re: Continuous bijection (not necessarily homeo)

On Jun 4, 8:51 am, Harun Al-Rashid <ab...@xxxxxxxxxxx> wrote:
Hi! My problem is:

Show that there is no continuous bijection from {z in C | |z| < 2}
to {z in C | |z| < 2 and z is not in [-1,1]}.

(here the real interval [-1,1] is considered as a subset of C)

TIA

The theorem of Invariance of Domain tells us that an
injective continuous map f : U --> R^2 from an open
subset U of R^2 to R^2 is an homeomorphism U --> f(U).

Thus any continuous injection from {z in C | |z| < 2}
to {z in C | |z| < 2 and z is not in [-1,1]} would be
an homeo, but the two spaces are not homeomorphic
(for example, one is contractible and the other is not)

-- m
.