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Therefore, the topologist's sine curve is always compact, hence locally compact. But in the article, it says: "T is the continuous image of a locally compact space (namely, let V be the space {−1} ∪ (0, 1], and use the map f from V to T defined by f(−1) = (0,0) and f(x) = (x, sin(1/x)) for x :
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I guess that some of the statements in this article refer to another sort of sine curve, where you just add (0,0) to the graph of sin(1/x). Then it would make sense to take the closure of it and then it would not be locally compact, but the image of a locally compact set (even a compact set)
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Later, it says in the article, that you may a variation, named "closed topologist's sine curve", which is now exactly the closure of the graph and therefore - by defintion - equal to the topologist's sine curve. So, the original topologist's sine curve is already the closed one...
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In the article, it says: You take the closure of the graph of sin(1/x) with x\in ]0,1]. The function is bounded, the domain is bounded, hence the graph is bounded. The closure of a bounded set w.r.t. the topology of a finite dimensional euclidean space is always compact. =:
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I'm not sure how to do this, but it would be neat to come up with a graph whose color becomes darker and/or more saturated to suggest increasing density approaching 0. As it is, at a certain point it just turns into a solid block.
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I have thus reverted the change: I think both definitions are used in the literature, so it does not really matter which one we give first, but we should be consistent in the article.
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An anonymous contributor in the last month changed the formula of the definition of the set T to include not just the origin but the whole interval
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The question is now: When topologists talk about "topologist's sine curve" do they mean the one with the interval or the one with just a point? --
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Image looks inconsistant with definition as the plot goes to negative x and in definition we take x from (0;1]. I may be just wrong though.
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This page is useless without the figure. Could I scan it from a book? I guess a mathematical figure cannot be copyrighted.
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My title is very generic. If we make the page we then have to update the title. But the page should be created!
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Seems like this space is too weird for
Iceweasel 2.0.0.3 (Debian version of Firefox) ;) Works fine in Konqueror.
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The current terminology of the article (with 'topologist's sine curve' denoting a non-closed set) is that of
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Knowledge. If you would like to participate, please visit the project page, where you can join
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Fixed. I don't know the answer to your last question. In
Munkres, the closure is used. –
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