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**.
\nToday, 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.<\/p>\n
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. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[65],"tags":[],"class_list":["post-156","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-2w","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/156","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=156"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/156\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=156"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=156"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=156"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nLet’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:
\n1. S ⊂ T
\n2. For each open set Os<\/sub> ∈ (S,σ), ∃ open set OT<\/sub> ∈ (T,τ): Os<\/sub> = OT<\/sub> ∩ S.
\nSo 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.
\nUsing 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.
\nAnother 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*.
\nWe 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.
\nFor 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).).
\n———
\nSince 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.
\nTo 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.
\nUsing 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.
\nEach 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.
\nSo – 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.<\/p>\n","protected":false},"excerpt":{"rendered":"