- #1

- 1,569

- 2

## Main Question or Discussion Point

i'm trying to define what it would mean for a set to be continuous.

what i'd like to say is that S is continuous if it is homeomorphic to [0,1], (0,1], or (0,1). (perhaps that's redundant already?)

but i'm not sure if that captures all the sets i'd like to think of as continuous. my main dilema is whether or not [0,1]x[0,1] would be continuous. is [0,1]^2 homeomorphic to [0,1]? i know there is a continuous "space filling curve" that maps [0,1] onto [0,1]^2 but i'm not sure it has a continuous inverse. does it?

if not, then the definition would be that S is continuous if it is homeomorphic to I^n, where I is some interval and n is a cardinal number. in this definition, the infinite dimensional hypercube [0,1]^[0,1] is included so that infinite dimensional manifolds could be classified as continuous or not.

what i'd like to say is that S is continuous if it is homeomorphic to [0,1], (0,1], or (0,1). (perhaps that's redundant already?)

but i'm not sure if that captures all the sets i'd like to think of as continuous. my main dilema is whether or not [0,1]x[0,1] would be continuous. is [0,1]^2 homeomorphic to [0,1]? i know there is a continuous "space filling curve" that maps [0,1] onto [0,1]^2 but i'm not sure it has a continuous inverse. does it?

if not, then the definition would be that S is continuous if it is homeomorphic to I^n, where I is some interval and n is a cardinal number. in this definition, the infinite dimensional hypercube [0,1]^[0,1] is included so that infinite dimensional manifolds could be classified as continuous or not.