Latest revision as of 22:58, 22 February 2017

Note: not to be confused with Homomorphism which is a categorical construct.

Definition

If $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ are topological spaces a homeomorphism from $X$ to $Y$ is a^[1]:

• Bijective map, $f:X\rightarrow Y$ where both $f$ and $f^{-1}$ (the inverse function) are continuous

We may then say that $X$ and $Y$ (or $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ if the topology isn't obvious) are homeomorphic^[1] or topologically equivalent^[1], we write this as:

• $X\cong Y$ (or indeed $(X,\mathcal{J})\cong(Y,\mathcal{K})$ if the topologies are not implicit)

  Note: some authors^[1] use $\approx$ instead of $\cong$^{[Note 1]} I recommend you use $\cong$.

Claim 1: $\cong$ is an equivalence relation on topological spaces.

Global topological properties are precisely those properties of topological spaces preserved by homeomorphism.

Notes

1. Jump up ↑ I recommend $\cong$ although I admit it doesn't matter which you use as long as it isn't $\simeq$ (which is typically used for isomorphic spaces) as that notation is used almost universally for homotopy equivalence. I prefer $\cong$ as $\cong$ looks stronger than $\simeq$, and $\approx$ is the symbol for approximation, there is no approximation here. If you have a bijection, and both directions are continuous, the spaces are in no real way distinguishable.

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2 1.3 1.4} Introduction to Topological Manifolds - John M. Lee

OLD PAGE

Not to be confused with Homomorphism

Homeomorphism of metric spaces

Given two metric spaces $(X,d)$ and $(Y,d')$ they are said to be homeomorphic^[1] if:

• There exists a mapping $f:(X,d)\rightarrow(Y,d')$ such that:
  1. $f$ is bijective
  2. $f$ is continuous
  3. $f^{-1}$ is also a continuous map

Then $(X,d)$ and $(Y,d')$ are homeomorphic and we may write $(X,d)\cong(Y,d')$ or simply (as Mathematicians are lazy) $X\cong Y$ if the metrics are obvious

TODO: Find reference for use of $\cong$ notation

Topological Homeomorphism

A topological homeomorphism is bijective map between two topological spaces

1. $$f$$ is bijective
2. $$f$$ is continuous
3. $$f^{-1}$$ is continuous

Technicalities

This section contains pedantry. The reader should be aware of it, but not concerned by not considering it In order for $f^{-1}$ to exist, $f$ must be bijective. So the definition need only require^[2]:

1. $f$ be continuous
2. $f^{-1}$ exists and is continuous.

Agreement with metric definition

Using Continuity definitions are equivalent it is easily seen that the metric space definition implies the topological definition. That is to say:

• If $f$ is a (metric) homeomorphism then is is also a topological one (when the topologies considered are those those induced by the metric.

Terminology and notation

If there exists a homeomorphism between two spaces, $X$ and $Y$ we say^[2]:

• $X$ and $Y$ are homeomorphic

The notations used (with most common first) are:

1. (Find ref for $\cong$)
2. $\approx$^[2] - NOTE: really rare, I've only ever seen this used to denote homeomorphism in this one book.

See also

• Composition of continuous maps is continuous
• Diffeomorphism

References

1. Jump up ↑ Functional Analysis - George Bachman Lawrence Narici
2. ↑ ^{Jump up to: 2.0 2.1 2.2} Fundamentals of Algebraic Topology, Steven H. Weintraub