how to show that an open disc that is inside of any circle n

how to show that an open disc that is inside of any circle (no matter how small) is homeomorphic to a punctured sphere?

Solution

Two topological spaces X and Y are said to be homeomorphic if there are
continuous map f : X Y and g : Y X such that
f g = IY and g f = IX.
Moreover, the maps f and g are homeomorphisms and are inverses of each other, so we may
write f 1 in place of g and g1 in place of f.
Here, IX and IY denote the identity maps.
When inspecting the definition of homeomorphism, it is noted that the map is required to be
continuous. This means that points that are “close together” (or within an neighborhood, if a
metric is used) in the first topological space ie disc are mapped to points that are also “close together”
in the second topological space ie punctured sphare. Similar observation applies for points that are far apart.
As a final note, the homeomorphism forms an equivalence relation on the class of all topological spaces.
• Reflexivity: X is homeomorphic to X.
• Symmetry: If X is homeomorphic to Y , then Y is homeomorphic to X.
• Transitivity: If X is homeomorphic to Y , and Y is homeomorphic to Z, then X is homeomorphic to Z.
The resulting equivalence classes are called homeomorphism classes.