[Math] Union and Intersection of sets

elementary-set-theory

I was reading Introduction to Topology by George L. Cain and found myself struggling with this definition mentioned in the book.

Let $X$ be a set, and suppose $C$ is a collection of subsets of $X$.
Then if $C$ = Empty Set, Union of $C$ is an Empty Set too and Intersection of $C$ is the set $X$.

Now my questions are:

  1. If $C$ is an Empty Set and also the collection of subsets of $X$ then isn't it true that $X$ is also essentially an empty set. For example, let $X = \{1,2\}$ then according to the definition $C = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}$.
    So, in the same manner $C$ can be an Empty Set only when the collection of subsets of $X$ is an Empty Set or $X = \emptyset$.

  2. Union of $C$ is the union of all the elements present in $C$. So if $C$ is an Empty Set, then the only set present in the collection of sets $C$ is the value of an Empty Set.
    Here lies my question: We know how to find the Union of $2$ or more sets but with respect to what should I find the Union of $1$ set? Also, at the back of my mind I know its an Empty set because there aren't any other sets present in $C$ but how do I know for sure?

  3. Intersection of $C$ is the intersection of all the elements present in $C$. So if $C$ is an empty set, again I present the same question, with respect to what should I take the intersection? Also, generally if $C$ was not an empty set and would be something like $C = \{\emptyset, \{1\}\}$, the intersection would be equal to an Empty Set as the set $\{1\}$ has subsets $\{1\}, \emptyset$.
    So how exactly does the Intersection of $C$ when $C$ is an Empty Set return set $X$?

Best Answer

Intuitively, when the collection $\mathscr C$ grows larger, its union grows larger and its intersection grows smaller. Going the other way, when the collection $\mathscr C$ grows smaller, its union grows smaller and its intersection grows larger. Taken to the extreme, when the collection $\mathscr C$ is as small as possible (empty), its union is as small as possible (empty) and its intersection is as large as possible (the whole space).

More technically, a point $x\in X$ is in $\bigcup \mathscr C$ if there is a member $C$ of the collection $\mathscr C$ with $x\in C$. When $\mathscr C$ is empty, this can't happen, so no point qualifies to be in $\bigcup \mathscr C$.

Similarly, a point $x\in X$ is in $\bigcap \mathscr C$ if $x\in C$ for every member $C$ of the collection $\mathscr C$. When $\mathscr C$ is empty, this is vacuously true (you can't demonstrate a member of $\mathscr C$ that fails to contain $x$).

So, the union of a family "starts out empty" and grows as you add sets to the family (more points qualify to belong), and the intersection of a family "starts out universal" and shrinks as you add sets to the family (fewer points qualify to belong).

Edit: I can't comment any more or the comments will be moved to chat; so I will "cheat" and comment here (sorry for the breach of protocol). No, it doesn't go beyond $X$ itself because we specified that $\mathscr C$ was a collection of "subsets of $X$" to begin with. I know that's a little vague, but $X$ is the universal set in this context.

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