Is a topological space considered to be a class in set theory?

Question

Question: Is a topological space considered to be a class in set theory?

So, by the might of wikipedia, I've found that

...a topological space may be defined as a set of points, along with a set of neighbourhoods for each point...

This sounds like a set of sets, because for every point element there is a neighborhood set. Does this make it a class in set theory, as shown here:

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

Then in order to understand what a class is, via ncatlab, I found what a collection is here:

use the word collection to denote a bunch of “things”

Which sounds self-referential. It seems like that the idea of an element and set are fairly concrete, and then a class/collection is just a bag that you throw things in.

On a slight tangent, how do you define higher level classes in set theory? Is there any notation that can show that this is a class of a class?

Answer 1

A topological space is an ordered pair $(X, \mathcal T) $ where $X$ is a set and $\mathcal T$ is a set of subsets of $X$ satisfying certain axioms. An ordered pair of sets is itself a set. It is a class in the sense that every set is a class, but it has no special reason to be called a class instead of a set.

Answer 2

Sets can contain sets. In the usual system of set theory, everything is a set - we start with the emptyset and "build up" (for a technical statement and proof of this, see e.g. here). For instance, we identify $0$ with $\emptyset$, $1$ with $\{\emptyset\}$, $2$ with $\{\emptyset,\{\emptyset\}\}$, and so on. Similarly, ordered pairs are defined purely set theoretically. So no, a topological space is still a set.

Classes enter the picture when we want to talk about collections - or if you prefer, properties - which cannot correspond to sets. For example, there cannot be a set of all sets not containing themselves - put another way, the collection of sets which don't contain themselves is a proper class. (Every set is a class but not conversely; we use the term "proper class" to refer to a class which is not a set.)

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There are "higher set theories" which treat sets and classes - for instance, NBG. They have their own analogues of "non-set collections," generally called hyperclasses, and so on - Russell's paradox tells us that however far we go we'll always have reasonably-definable properties which don't correspond to objects in our framework.

Answer 3

I believe your question has been adequately answered, but I will iterate: A topological space is a pair consisting of a set of points and a subset of the powerset of the set of points with the property that each element of the set of points is contained in at least one element of this subset. (I.e., each point is present in at least one subset of the set of points that is deemed "open".) So the two members of the pair of a topological space are both classes, but only in the boring sense: all sets are classes.

It is worth pointing out that the set of "points" need not be points in what is normally considered a space. There is a surprising topological proof of the infinitude of primes where the topological space is the set of integers and the open sets are congruence classes of integers modulo various integers. One difficulty with your idea of specifying the "depth" of a set is that the integers, in their usual set theoretic model, has infinite depth, since incrementing an integer increases the depth by one. So as soon as you have the integers, you have infinite depth and then when you want to talk about actually large sets, annotating set depth becomes useless visual clutter.

In comments to another post, you object to defining classes via properties. Perhaps it would help if you understood how that is different from a thing being a set. A thing is a set if is constructed according to the axioms of your set theory. In, for instance ZF (abbreviating mightily), you get the empty set, an infinite set, and a few (fairly familiar ways) to construct new sets out of old sets. Only those things are sets.

A class is given by a decision procedure that, given a thing, determines whether that thing is a member of the class. All sets are classes because the procedure "verify that this is a member of this set" produces a class that is identical to the set. However, there are decision procedures that do not produce sets because the object they produce cannot be constructed by the axioms. Consequently, there are classes that are not sets. Then, the class is equivalent to its decision procedure -- i.e., each class is equivalent to the property expressed by its decision procedure. So a class, its decision procedure, and the property it encodes are all equivalent.