Topological equivalence

An equivalence relation between topological spaces. Two topological spaces $X$ and $Y$ are said to be topologically equivalent (or homeomorphic), if there exists a homeomorphism, continuous map between the spaces, $H\in C^0(X,Y)$ which has a continuous inverse $H^{-1}\in C^0(Y,X)$. Topological equivalence is a reflexive, symmetric and transitive binary relation on the class of all topological spaces.

Topologically equivalent spaces are indistinguishable from the point of view of any property which is purely topological (i.e., is formulated in terms of the behavior of open/closed sets).

Topological equivalence of spaces with additional structures

The term "topological equivalence" by extension is used for additional structures on topological spaces. For instance, if $A\subseteq X$ and $B\subseteq Y$ are two subspaces of the topological spaces (considered together with the corresponding embeddings $i_A:A\to X$, $i_B:B\to Y$), then the topological equivalence of such subspaces is a pair of homeomorphisms $h:A\to B$ and $H:X\to Y$ such that $$ H\circ i_A=i_B\circ h,\qquad i_A:A\to X,\ i_B:B\to Y. $$ If $X,Y$ are smooth manifolds and $A,B$ their submanifolds, (e.g., embedded closed smooth curves), then this equivalence describes the way how the submanifolds are embedded in the ambient manifolds. For instance, a meridian on a 2-torus is not topologically equivalent to a contractible loop.

Topological equivalence of partitions

More generally, if $X=\bigsqcup_{\alpha\in A} X_\alpha$ and $Y=\bigsqcup_{\beta\in B} Y_\beta$ are two partitions of the spaces $X,Y$ into the disjoint unions of subsets indexed by two sets $A,B$, then the topological equivalence of such partitions means that there is a homeomorphism $H:X\to Y$ and a bijection $h:A\to B$ such that $H(X_\alpha)=Y_{h(\alpha)}$, that is, which sends (necessarily in a one-to-one way) the subsets of the partitions into each other.

This quite general scheme is in fact a common roof for several notions arising in different areas.

Equivalence of the group actions and dynamical systems

Let $G$ be an abstract group acting on two spaces $X,Y$: there are well defined applications $G\times X\to X$ and $G\times Y\to Y$. Each action defines the partition of $X$, resp., $Y$ into orbits of action, $$ X=\bigsqcup_x G(x),\quad Y=\bigsqcup_y G(y),\qquad G(x)=\{g\cdot x:~g\in G\},\quad G(y)=\{g\cdot y:~g\in G\}. $$ The two actions are called (orbitally) topologically equivalent, if there exists a homeomorphism $H$ preserving these partitions, i.e., sending orbits into orbits.

Example. Let $X,Y$ be two smooth manifolds and $v,w$ two vector fields on them. Each vector field defines the flow which is an action of the real line $\R$ (considered as the additive group). Two vector fields are called topologically orbitally conjugate, if the corresponding partitions of $X$ (resp., $Y$) into phase trajectories are homeomorphic.

A more restrictive notion of the conjugacy appears when the conjugating homeomorphism is required to conjugate also the action of $G$ on each orbit. This amounts to the requirement that $$ H(g\cdot x)=g\cdot H(x),\qquad \forall x\in X,\ \forall g\in G. $$

Example. Two diffeomorphisms $f\in\operatorname{Diff}(X)$, $g\in\operatorname{Diff}(Y)$ define the actions of the group $\Z$ on the respective manifolds. The topological equivalence of the corresponding actions means that there exists a homeomorphism $H:X\to Y$ which conjugates $f$ with $g$: $H\circ f=g\circ H$.

One may generalize slightly this construction by assuming that one may also twist the group $G$ by a suitable automorphism: two actions are called topologically equivalent, if there exists a homeomorphism $H:X\to Y$ and an automorphism $\varphi\in\operatorname{Aut}(G)$ such that $$ H(g\cdot x)=\varphi(g)\cdot H(x),\qquad \forall x\in X,\ \forall g\in G. $$

These two examples provide a convenient definition for the topological equivalence of dynamical systems.

Motivations

Topological equivalence in many situations is the weakest which leaves hopes that the list of equivalence classes is finite. For instance, the topological classification of linear operators on $X=\R^n$ is finite, whereas a linear (or just $C^1$-smooth) equivalence cannot change the eigenvalues and hence has uncountably many different classes. The most famous finite topological classification is for nondegenerate singular points of planar vector fields into the three Poincare types (node/focus, saddle, center).

However, even the topological equivalence in some situations does not guarantee that the number of the equivalence classes is finite. For instance, diffeomorphisms of the circle $\mathbb S^1$ into itself have a real topological invariant, the rotation number.

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References

[a1]	R. Engelking, "General topology" , Heldermann (1989)