From: renfrod@central.edu (Dave L. Renfro) Subject: Re: First-Countable, Second-Countable, HUH? Date: 15 Apr 2001 15:35:04 -0400 Newsgroups: sci.math Summary: Comparison of countability axioms in topology Euclid's Dog [sci.math Sun, 15 Apr 2001 03:27:26 GMT] wrote > Given a countable set X that is also first-countable, I'd like I assume "set" means "topological space"? > to show that it is second-countable. I put together what looked > like a convincing enough proof, but I ended up not needing the > fact that the set X was first-countable (kind of like putting > together a bicycle and having a single nut left over). Why then > do I need that for the proof? All I could figure is that it is > a roundabout way to say that the set X has a local base at each > point x in X. The fact that every point has a local base should be something proved right after the concepts were introduced. This is always true. But this isn't what first countable means. First countable means that every point has a COUNTABLE local base. > My proof goes something like this: > > Let X be a countable space and U an open set in X. For each x > in U, U is a neighborhood of x. Since X is first-countable, > x has a local base, As I mentioned above, this doesn't require first-countability. For each x in X let U(x) be the collection of all neighborhoods of x. Then U(x) will satisfy what I assume you mean by local base at x. [Certain subcollections of U(x) will also be a local base at x, unless you're dealing with a really trivial space.] > so we can find an open set V containing x, > such that x in V in U. We can find such a V for every x in U, > and the union of all such open sets V is also open and "equal" > to U. Since there are only countable many x, there are countable > many V and X is second-countable. Here's an analogy that might help to show where your argument breaks down: Each semi-open interval (-oo, r) in the real line for any real number r is the union of all sets V of the form (-oo, x) for x a rational number less than r. Since there are only countably many x's, there are countably many V's, and hence countably many sets (-oo, r). [Don't tell Ross A. Finlayson about this.] > See? I used the fact that X was first-countable only to claim > that each x had a local base. I really didn't need that it was > (or wasn't) first-countable. The fact that the number of > elements in X (and any subset thereof) is countable, gave me > the second-countable part. > > I know I missed something. Can you help? Thanks in advance... A first countable space need not be second countable --- consider any uncountable discrete space. A countable space need not be first countable. In fact, there exist countable spaces that have no points of first countability. For some examples, see --->>> Peter J. Harley, "A countable nowhere first countable Hausdorff space", Canadian Math. Bull. 16 (1973), 441-442. Richard Willmott, "Countable yet nowhere first countable", Mathematics Magazine 52 (1979), 26-27. In fact, Leslie Foged constructed 2^c nonhomeomorphic countable spaces having no points of first countability in his 1979 Ph.D. Dissertation "Weak Bases for Topological Spaces" under Ron Freiwald at Washington University. As for what you want to prove, namely that for countable topological spaces, first countability implies second countability, you want to be a bit more explicit in your proof. First, tell your reader exactly which countable collection of open sets you're going to show is a base. [Each point has a countable local base (these bases may not be unique, so make sure you don't refer to any of them as "the local base at x" until you've fixed a choice of them in your proof), there are countably many points, a countable union of countable sets is countable, . . .] Having done this, now prove this collection is a base for the topology on X. Look up the definition of a base and make sure that you cover each part of it (pun originally accidental, but I'll take credit for it). If you really want to impress sci.math readers, write an essay explaining how this result can be incorporated into logic based probability using the proximity function. [See David C. Ullrich's April 13, 2001 post at for an excellent example along these lines.] Dave L. Renfro