I am reading a topology definition:

Let $X$ be a set and let $\tau$ be a family of subsets of $X$. Then $\tau$ is called a topology on $X$ if:

• Both the empty set and $X$ are elements of $\tau$
• Any union of elements of $\tau$ is an element of $\tau$
• Any intersection of finitely many elements of $\tau$ is an element of $\tau$

If $\tau$ is a topology on $X$, then the pair $(X, \tau)$ is called a topological space. The notation $X_\tau$ may be used to denote a set $X$ endowed with the particular topology $\tau$. The members of $\tau$ are called open sets in $X$.

My question is:

Are the members of $\tau$ open sets or only called open sets?

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The elements in $\,tau\,$ are defined to be the open sets in that topology. Nothing is naturally open or closed, it is just a matter of definitions...which, of course, must make sense with axioms, other definitions and etc. – DonAntonio Aug 23 '12 at 17:06
At the risk of sounding silly, what's the difference between being an open set, and merely being called one? – Ben Millwood Aug 23 '12 at 17:22

This definition abstracts the idea of an open set in a metric space, so in a sense the members of $\tau$ are only called open sets - they aren't open in the sense that they contain a ball around every point, because there isn't any distance to say what that means. (Of course, as other people have pointed out, once you decide to call them open sets, the distinction between being called open and actually being open is somewhat unclear).

The reason this definition is chosen is that the collection of open sets in a metric space (defined in the usual metric spaces way) forms a topology in this sense. So in that case the sets in $\tau$ are open both because they are in $\tau$ and because they are open in the metric sense.

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So why are the open sets in a metric space really, really open? – Michael Greinecker Aug 23 '12 at 17:20
My assumption was that the OP was introduced to topology in a similar way to me, and learnt "open" in the sense of metric spaces first. Then it's natural to think of this as somehow being the "real" definition of open, and the language "the members of $\tau$ are called open sets" seems suspicious. Hence I have inserted words into the question in my head to reflect this, to get: "are open sets in a topology open in the sense I already know, or do we call them that because they behave similarly?". I hope this addresses what you were getting at. – Matthew Pressland Aug 23 '12 at 17:33
I've reworded to try to make this a little clearer in the answer. – Matthew Pressland Aug 23 '12 at 17:40

We call these barking, furry animals dogs. But are they really dogs or are they just called this way? Actually, since my native language is German, I would propose that they are not really dogs. They are really Hunde, it's just that they are called dogs in English.

This kind of reasoning doesn't become less absurd when being applied to mathematics. Actually, in mathematics you basically define things by calling certain things a certain way. Mathemticians are firmly on the side of Humpty Dumpty:

"I don't know what you mean by 'glory,' " Alice said. Humpty Dumpty smiled contemptuously. "Of course you don't—till I tell you. I meant 'there's a nice knock-down argument for you!' " "But 'glory' doesn't mean 'a nice knock-down argument'," Alice objected. "When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean—neither more nor less." -- Lewis Carroll, Through the Looking-Glass

So we are actually not lying when we call an open set open.

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