Is a topology a set or a family?

Question

So I've just started learning topology and a lot of the definitions confuse me. My main problem is that some of the definitions are quite inconsistent to me for example in topology without tears the author says that a topology $\tau$ is a set of subsets of $X$ and then he proceeds with the axioms. However I've also seen a lot from other sources that a topology is a family of subsets of $X$ and then they proceeds to describe the same axioms. However they do not refer to the topology $\tau$ as being a set at all which confuses me.

I think my confusion is the definition on what a family actually is ive seen lots of confusing definitions on what a family is like it is a surjective function however I cannot seem to grasp the idea.

So why do they define them differently and what is a family?

Thanks in advance.

Answer 1

I think the word "family" is somewhat ambiguous. For example, Wikipedia says

In set theory and related branches of mathematics, a collection $F$ of subsets of a given set $S$ is called a family of subsets of $S$, or a family of sets over $S$. More generally, a collection of any sets whatsoever is called a family of sets or a set-family or a set-system.
The term "collection" is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class rather than a set.

Here allowed to contain repeated copies of any given member means what Wikipedia denotes as an indexed family :

More formally, an indexed family is a mathematical function together with its domain $I$ and image $P$. Often the elements of the set $P$ are referred to as making up the family. In this view indexed families are interpreted as collections instead of as functions. The set $I$ is called the index (set) of the family, and $P$ is the indexed set.

The first quotation shows that a family of sets can be understood either as a set of sets or as function $f : I \to P$ where $P$ is a set of sets. This is indeed vague and leaves much scope for interpretation. The same vagueness of notation can be found in many textbooks.

For a given set $X$ we may consider

• sets $\tau$ of subsets of $X$, i.e. subsets of $\tau \subset \mathfrak P(X)$ = power set of $X$.

• "indexed collections" of subsets of $X$, i.e. functions $\theta : I \to \mathfrak P(X)$.

Each subset $\tau \subset \mathfrak P(X)$ can be canonically identified with the inclusion function $\iota(\tau) : \tau \hookrightarrow \mathfrak P(X)$; this produces a "self-indexed collection" of subsets of $X$. Conversely, each function $\theta : I \to \mathfrak P(X)$ determines the subset $\text{im}(\theta) = \theta(I) \subset \mathfrak P(X)$. Clearly $\text{im}(\iota(\tau)) = \tau$, but in general $\iota(\text{im}(\theta)) \ne \theta$. In fact, for a given $\tau \subset \mathfrak P(X)$ there are many $\theta : I \to \mathfrak P(X)$ such that $\text{im}(\theta) = \tau$. One can even show that the "collection" of all $\theta : I \to \mathfrak P(X)$ such that $\text{im}(\theta) = \tau$ is not even a set, but a proper class.

The standard is to define a topology on a set $X$ as a set of subsets of $X$ satisfying suitable axioms.

If you regard the word family as a synonym for set, then you do not get conflicting definitions. If you regard a family as a function, then a topology on $X$ would be some function $\theta : I \to \mathfrak P(X)$ from an index set $I$ to the power set of $X$ which is often written in the form $\{U_i\}_{i \in I}$ (indexed collection of subsets). You can easily modify the axioms for a topology to obtain similar axioms for an indexed collection of subsets of $X$. It is fairly obvious that $\theta$ satisfies these modified axioms if and ony if $\text{im}(\theta)$ is a topology in the standard sense. The essential disadvantage of the alternative definition is that indices are completely unnessary - many formally distinct families of "indexed topologies" give the same "standard topology".

Answer 2

"A set of sets" is a somewhat abstract notion. It is, formally, what a topology is, but people also need to know how to think about them. Having an informal "hierarchy" of container names gives newcomers some mental intuitive help (or at least, that's the idea).

Thus we get "a family of sets".