Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this?

I’m assuming this has something to do with the following observation: \cap_{n=1}^{\infty} (-\frac{1}{n},\frac{1}{n}) = \{0\} and there is some reason we don’t want singletons to be considered open, I am wondering what this reason is. Am I thinking in the right direction here?

Thanks 🙂

Answer

You need to think about what the intuition behind open sets are. One way to think about it is through neighborhoods: an open set is a set which is a neighborhood of each of its points. What is a neighborhood of a point? A neighborhood of a point x is a set that contains of all points that are “sufficiently close” to x (what does “sufficiently close” mean? It depends on the situation; you think of different neighborhoods perhaps specifying different degrees of closeness). In particular, any set that contains a neighborhood of x is itself a neighborhood of x. And specifying two degrees of closeness specifies another degree of closeness that makes sense (the smaller of the two at any given place, say).

So: if you think about open sets as sets that are neighborhoods of all of the points they contain, then it is natural that the union of any family of pen sets will be open: each point in the union is one of the open sets, and that open set is a neighborhood, and the union contains that neighborhood and so is itself a neighborhood. So the arbitrary union of open sets should still be open.

What about intersection? Well, if you take two open sets O_1 and O_2, and you consider a point x in O_1\cap O_2, then O_1 contains all points that are 1-sufficiently close to x, and O_2 contains all points that are 2-sufficiently close to x (with “1-sufficiently” and “2-sufficiently” describing the two degrees of closeness required), so O_1\cap O_2 will contains all points that are both 1-sufficiently and 2-sufficiently close to x, and so it contains all points that are “sufficiently close” to x for some meaning of “sufficiently close”, so it is also an open set. This gives you, inductively, any finite intersection.

But what about arbitrary intersections? Then you run into trouble, because specfying two degrees of “closeness” gives you a degree of closeness (the smaller one), but an infinite number of degrees of closeness may end up excluding everything! (Just as in your example, taking the intersection of all (-\frac{1}{n},\frac{1}{n}), which specify all points that are \frac{1}{n}-close to 0, but the intersection excludes everything). So we don’t want to require that an arbitrary intersection of neighborhoods be a neighborhood, and so we don’t want to require that an arbitrary intersection of open sets be an open set.

Attribution
Source : Link , Question Author : WWright , Answer Author : Arturo Magidin

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