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Non-metrizable Hausdorff space

I've been reading all this stuff about how Hausdorff spaces are a great generalization of metric spaces that extends the concepts of limits and stuff, but every Hausdorff space I can think of has a natural metric structure, so I can't really see what the fuss is about. Can anyone give me a simple, easy-to-understand example of a Hausdorff space which is not metrizable? —Keenan Pepper 09:30, 5 August 2006 (UTC)

Personally, I like the Long line (topology). Melchoir 17:08, 5 August 2006 (UTC)

Oh, and speaking of order topologies, I can't imagine that the ordered plane could be metrizable. Melchoir 17:36, 5 August 2006 (UTC)

Okay, I'm still trying to grasp what the long line is, exactly. Is $\omega _{1}$ the order type of [0,1)? —Keenan Pepper 09:21, 6 August 2006 (UTC)

I trust that others will correct me if necessary, but I think that the long line (to be more precise, the long ray which is half of the long line) is $\omega _{1}$ copies of [0,1). As a set, it is (0,1) × [0,1). but it has a different topology.

How about C(R), the space of continuous functions on the reals; is that Hausdroff and not metrizable? -- Jitse Niesen (talk) 11:18, 6 August 2006 (UTC)

I'm not sure, but I'd suspect that it is metrisable, by analogy with C[a,b] - if you look at continuous functions on a finite interval you can define a metric based on an integral norm, so while you can't exactly do that (given that most of your integrals will be infinite), I suspect you can probably induce some kind of metric on it. Confusing Manifestation 00:44, 9 August 2006 (UTC)

Every set is metrizable (just use the discrete metric). But C(R) with its limit topology is certainly not metrizable, but is Hausdorff. A locally convex TVS is metrizable iff its topology can be given by a countable family of seminorms. The seminorms for C(R) are indexed by no fewer than uncountably many of the compact subsets of R. -lethe ^{talk +} 20:19, 10 August 2006 (UTC)

When we talk about metrizability, we need to know which topology is used. For C(R), (to the best of my memory without looking up) the compact open topology is metrizable but the topology of pointwise convergence is NOT. The test space on R is not metrizable. Twma 05:46, 9 August 2006 (UTC)

On Limits

Is there an algorithm with a finite number of steps that will allow you to find the limit of any function, provided the limit exists? This question has bothered me for a bit. Phrased differently, is there a universal way to find a limit when the limit exists? Thanks AmateurThinker 15:47, 5 August 2006 (UTC)

L'Hôpital's_rule? doubt it works every time though.

Yes. see Limit of a function or any good (pre)calculus text book. Remember that L'H%C3%B4pital's rule only applies when the limit is an indeterminate form. 48v 17:27, 5 August 2006 (UTC)

I'd say the answer is: No, no such algorithm exists for the general case. There may be algorithms that work for specific, possibly large, classes of functions, but they will not be fully general. --Lambiam^Talk 17:57, 5 August 2006 (UTC)

I would argue that an algorithm can inclue logic statements and cases, thus you could concive of a 'master agorithm' bringing together the cases for classes of functions. This is, indeed, the algorithm that people use to find a limit. 48v 19:48, 5 August 2006 (UTC)

Where did you get the idea that every function belongs to some class for which there is an algorithm? -- Meni Rosenfeld (talk) 21:36, 5 August 2006 (UTC)

As I understood it, AmateurThinker was asking only about functions for which a limit exists. It is not clear to me that limits exist in this 'busy beaver' function. Perhaps I am overlooking something, but I have not found one yet which cannot be found using an algorithm of some fashion. 48v 01:00, 6 August 2006 (UTC)

Indeed. See the busy beaver function. --Zemyla^t 23:26, 5 August 2006 (UTC)

This is a naïve question, because we cannot even write down a finite expression to define most functions φ: R→R. (We have only countably many finite definitions, but uncountably many functions.) However, if we suitably restrict the kind of functions we admit, we have possibilities. If you're up to it, have a look at Joris van der Hoeven's doctoral dissertation (start at page 165) for an extended discussion. Or, try this abbreviated discussion. --KSmrq^T 06:43, 6 August 2006 (UTC)

A place for this?

In this post on the Wizards of the Coast boards, a D&D 3.5 character (actually, three of them) can do approximately $\left(2.5\times 10^{36530}\right)\uparrow \uparrow 73600$, which is almost certainly a record for the largest number ever appearing in a role-playing game. So where could this go on Wikipedia? I noticed we have an article about Pun-Pun, which is from the same boards. --Zemyla^t 23:19, 5 August 2006 (UTC)

How can a character (or you) do a large number ? Enumerate its antecessors ? Write it ? Repeat a movement this large number of times ? Please explain before putting that in WP. --DLL 17:24, 6 August 2006 (UTC)

I don't think a post on a bulletin board can serve as a reputable source for a citation. --Lambiam^Talk 16:06, 7 August 2006 (UTC)