Question

In: Advanced Math

A function f : X ------> Y between two topological spaces ( X , TX )...

A function f : X ------> Y between two topological spaces ( X , T_X ) and ( Y , T_Y ) is called a homeomorphism it has the following properties:

a) f is a bijection (one - to- one and onto )

b) f is continuous

c) the inverse fucntion f ^-1 is continuous ( f is open mapping)

A function with these three properties is sometimes called bicontinuous . if such a function exists, we say X and Y are homeomorphic.

show that a complete metric space R is homeomorphic to the metric space ( 0 , 1 ), which is not complete . Metric defined as an ab solute value of the difference

Solutions

Expert Solution

Define a map

(a). f is one-one and onto:

one-one: for

and hence is one-one.

is onto: for take an arbitrary . Then, we find an such that . For,

Note that for ,

and hence is a number in Thus, there exists such that and hence is onto.

(b). is continuous: We know that is a continuous function and qutotient of two continuous function (with nonzero denominator ) is continuous .

(c). is also continuous: Note that for ,

which is continuous (since the log function is continuous and composition of continuous maps are continuous).

Hence and are homeomorphic .

But is not a complete metric space since is a Cauchy sequence in which does not converge in .

Colby Messinger answered 1 year ago

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