Definition of homeomorphic?

Question

I am looking up the definition of "homeomorphic" and the source I am looking at says there are two different definitions:

1. Possessing similarity of form,

2. Continuous, one-to-one, in surjection, and having a continuous inverse.

2. seems to be speaking of a particular function/mapping, so I'm okay with that. But "possessing similarity of form" is not rigorous so I don't understand what is meant by that. Does it just mean there exists a function that is continuous, one-to-one, in surjection, and has a continuous inverse from one set to another? Like when it is said that e.g. "the $2$-sphere is not homeomorphic to $\mathbb{R}^2$," does that mean there exists no function $f: S^2 \to \mathbb{R}^2$ such that $f$ is continuous, one-to-one, in surjection, and has a continuous inverse?

Answer 1

Two topological spaces are homeomorphic iff there exists a mapping as described in (2) between them, a continuous 1-1 map $f$ from one onto the other space whose inverse $f^{-1}$ is also continuous. Because of this condition, both $f$ and $f^{-1}$ are called homeomorphisms, i.e. maps that preserve the underlying topological structure of a space.

Thus the two topological spaces have the same topological structure. This gives a rigorous interpretation to (1); as the OP points out, without more context statement (1) about "similarity of form" lacks rigor.

The example of two spaces, the 2-sphere $S^2$ and the plane $\mathbb{R}^2$, that are not homeomorphic illustrates the topic. If these were homeomorphic spaces, their topological properties (preserved under homeomorphism) would be the same. But compactness is a topological property (preserved indeed by continuous surjections), and $S^2$ is compact but $\mathbb{R}^2$ is not compact. So we know these spaces are not homeomorphic.

Answer 2

The first definition is bizarre and is not at all standard mathematical terminology. Maybe "homeomorphic" can be used with that informal meaning in a non-mathematical context (though I haven't found any such usage by googling; the only other usage I can find is a different technical meaning in chemistry). In short, don't worry about that supposed first definition; you will never see anyone using it.

Answer 3

The source is likely referring to the English word "homeomorphic" and has nothing to do with mathematics. Notice how closely the definitions map to dictionary.com's homeomorphism:

noun

1. similarity in crystalline form but not necessarily in chemical composition.

2. Mathematics. a function between two topological spaces that is continuous, one-to-one, and onto, and the inverse of which is continuous.

From the etymology it seems that the chemical term "homeomorphous" dates back to 1832, so that usage effectively predates the field of topology itself (not counting the ever-prescient Euler).