Topological Subspaces

Just like you can define a sub-set of a set, or a sub-object of an object in a category, you can define a sub-*space* of a topological space. It's a pretty easy thing to understand; interestingly, a sub-space of a topological space works in pretty much exactly the same way as a sub-sets and sub-object. In fact, the topological definition of a sub-space is *identical* to the categorical definition of a sub-object when we're looking at the category of topologies, **Top**.

Today, I'm going to explain what a subspace is, and show you how the categorical sub-object corresponds to the topological subspace. Read on beneath the fold.

Let's start with the normal topological definition. Suppose we have two topological spaces, (T, τ), and (S,σ). (S,σ) is a sub-space of (T,τ) if and only if:

1. S ⊂ T
2. For each open set O_s ∈ (S,σ), ∃ open set O_T ∈ (T,τ): O_s = O_T ∩ S.

So saying (S,σ) is a subspace of (T,τ) means that S has a subset of the objects that are in T, and (S, σ) structures the objects that it contains in the same way as T; so (S,σ) preserves as much of the structure of (T,τ) as you can represent using the objects contained in S.

Using that definition, given a set of objects, S ⊂ T, we can construct a topological space *induced by* T by generating open sets using statement 2 from the definition of subspaces. We call the set of open sets σ the *relative topology* of T on S. The neighborhoods in the relative topological space (S,σ) are called the *relative neighborhoods* of S.

Another way of defining the relative neighborhoods without generating the induced space is to apply statement 2 from the definition of subspaces in the definition of relative neighborhoods: Suppose we have a subspace (S,σ) of a topological space (T,τ); and let o ∈ S. Then *N(o)* is a relative neighborhood of *a* if/f: ∃ a neighborhood *M(o)* in (T,τ) such that *N(o)= M(o) ∩ S*.

We can also define *relatively closed* sets; it's exactly the same trick of pulling statement two of the definition of subspaces into the definition of closed sets. Given a subspace (S,σ) of (T,τ), the *relatively closed* subsets of S are the sets C ⊂ S such that for *some* closed set D in (T,τ), C = D ∩ S.

For all subspaces (S,σ) of the topological space (T,τ), a function f : T → X is continuous if and only if f restricted to S is continuous on (S,σ). (f restricted to S is a function g : S → X;: ∀ s ∈ S, g(s) = f(s).).

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Since we'll use some category theoretical stuff in some later topology posts, let's take a look at how this corresponds to the categorical concept of sub-objects. The set of topological spaces forms a category **Top**, where the *objects* in Top are topological spaces; and the *arrows* between them are continuous functions.

To review a bit, in category theory, a sub-object is defined in terms of *monic arrows*. A monic arrow is the category theoretic version of an injective function: f: a → b is *monic* if/f for all other arrows g and h : x → a, fº g = f º h implies that g = h. In other words, a monic arrow will *only* map two other arrows to the same place if those two other arrows are the same.

Using monic arrows, we can define an *equivalence class* of arrows. Suppose we have two monic arrows f : b → a, and g : c → a. If there is an arrow h such that g º h = f, then f ≤ g. If f ≤ g and g ≤ f then f ≡ g.

Each equivalence class defines a set of sub-objects of a; and those sub-objects are treated as the same object, because with respect to a, they are indistinguishable.

So - let's think about this in the category of topological spaces. A *monic* arrow f : s → t, where s = (S,σ) and t = (T,τ) in the category of topological spaces is an *injective* function - that is, it maps every object in S to a distinct object in T. That is, every object in S can be identified with exactly one object in T. Further, there is a preservation of structure: all arrows that can be mapped through the sub-object s arrive at the same mapping in T; that is, if we have a continuous function from X to T, and we use it on S, it will always generate the same result as if we applied it to *T* directly. Since for the function to still be continuous on S (which it must be to be an arrow), that means that for all possible continuous functions that can be applied to S, S will behave exactly the same as T; preserving structure the same way that T does. So the open-set relationships will be preserved exactly as in the topological definition above.