Manifolds Unfolded

In Mathematics, while working with spaces i.e. sets equipped with some structure such as metric spaces, vector spaces, groups, rings etc., it is a general thing to ask whether the two spaces are identical with respect to the structure, for example, if we are dealing with vector spaces, we would like to ask whether two vector spaces are isomorphic, if we are working with sets, then we would like to ask if there is a bijection between the two sets, if we are dealing with groups (or rings), then we should inquire about whether they are group isomorphic (or ring isomorphic resp.). Thus, the general theme of studying spaces is to study maps that preserve the structure between the spaces, which are called morphisms.

Similarly in topology, it is natural to ask the question whether two topological spaces are ‘essentially the same’ or they are different. In this discussion, we shall try to make precise the idea of ‘being the same’ for two topological spaces (i.e. topologically the two spaces are indistinguishable). This would allow us to classify topological spaces.

The first step is to understand the suitable morphisms in this context. Topology is concerned with ‘neighborhoods’. A map that preserves neighborhoods will be the ideal candidate for a morphism between two topological spaces. It turns out that such maps precisely are the continuous maps between topological spaces.

Let us recall the definition of continuity at a point:

The above can be equivalently written as:

which says that continuity is synonymous to preserving neighborhoods. Let us now define what we mean by a continuous map between topological spaces.

As we said, since continuous maps preserve neighborhoods, they should preserve all the ‘topological aspects‘ of a space. In simpler words, a continuous transformation of a space into another, should not alter any feature of a space which is governed by its topology.
Hence, continuous maps are the right candidates for morphisms (structure preserving maps) between topological spaces.

Now let’s come to the question we are here for : When do we say two topological space are ‘identical’? The answer is when there is a bijection between the two spaces that preserves topological structure in both ways i.e. the forward and the inverse direction should preserve the ‘topological structure’. This leads us to the following definition:

We say that two topological spaces are homeomorphic, provided there exists a homeomorphism between the two spaces. We denote this relation as and we say that the spaces and are homeomorphic to each other.

Observe that, the relation of homeomorphism is an equivalence relation on any given collection of topological spaces. Thus, we can classify topological spaces into equivalence classes, which will be called homeomorphism classes: . Two spaces are in the same class if and only if they are homeomorphic to each other, i.e.

So now we have a formal understanding of homeomorphism and homeomorphic topological spaces, which are ‘identical‘ in all their ‘topological aspects’.

Let us discuss some examples and non-examples to get a clearer idea:

• Examples:
  • Consider the real line ℝ, it is homeomorphic to the interval. The map given by, which is a continuous bijection with the well-defined inverse function viz. (aka. the function) and continuous from . (†)

• Consider the real line ℝ, it is homeomorphic to the space (equipped with the subspace topology induced from the Euclidean topology on ). The homeomorphism in this case is given by the map sending . This map is a bijection and the inverse map is given by the restriction of the first coordinate projection map which sends . It is easy to check that both the maps and are continuous.

• Non-examples:
  • The real line ℝ and the closed bounded interval are not homeomorphic because the latter is compact, whereas the former is not. Had there been any homeomorphism from to ℝ, then ℝ would be a continuous image of a compact space. But as we know, continuous maps send compact spaces to compact ones. This clearly shows that the aforementioned two spaces arenot homeomorphic to each other.
    
  • The unit circle and the real line ℝ are not homeomorphic. If they were, then there would exist a homeomorphism . Then, after removing a point from the domain and the corresponding image point from the co-domain, and taking the restriction of , we would have again gotten a homeomorphism . But this is not the case as removing one point from the real line renders it disconnected with two components, but the circle minus a single point is remains connected. Also, we know: continuous maps take connected spaces to connected spaces, so there can be no homeomorphism f from onto ℝ.

Topological Property

A topological property (or topological invariant) is a property of a topological space that is invariant under homeomorphisms.

Examples: Connectedness, Compactness, First Countability, Second Countability, Separability etc. are all topological invariants.

Non-examples: Completeness, Boundedness, Total Boundedness etc. are not topological properties, although they are metric properties. For instance, the open interval is bounded, whereas the real line ℝ is not bounded; even though they are homeomorphic as we have seen in the first example (†).

Intuitively, we can think of homeomorphism in the following way : If we can transform one space continuously into another just by deforming, stretching, bending, twisting or compressing, but without any tearing or gluing, then and then only the two topological spaces are homeomorphic to each other. This method is visually handy while dealing with concrete topological spaces.

There is a joke saying: A topologist cannot distinguish between a coffee mug and a doughnut, for them, these two objects are identical ! (Guess why, because they are homeomorphic!)