Evandar's New Writeupshttp://everything2.com/?node=New%20Writeups%20Atom%20Feed&foruser=Evandar2001-07-12T01:28:48ZConnected (idea)http://m.everything2.com/user/Evandar/writeups/ConnectedEvandarhttp://m.everything2.com/user/Evandar2001-07-12T01:28:48Z2001-07-12T01:28:48ZIn <a href="/title/topology">topology</a>, an important <a href="/title/axiom">axiom</a> of <a href="/title/topological+space">topological space</a>s.
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<u><a href="/title/Definition">Definition</a></u> A topological space X is said to be <u>disconnected</u> if there are <a href="/title/nonempty">nonempty</a> subsets A,B of X such that AnB={}, AuB=X, and A and B are both open in X.
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Since A and B are <a href="/title/complement">complement</a>s in X, it follows that they must both be closed. So we could replace 'open' with 'closed' in the definition above. I will refer to A and B as a 'disconnection' of X (which isn't standard terminology). It's reasonably clear that the following is equivalent:
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<u><a href="/title/Lemma">Lemma</a></u> X is disconnected iff it has a proper, nonempty subset that is both open and closed.
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<u><a href="/title/Definition">Definition</a></u> A topological space X is <u>connected</u> if it is not disconnected. A subset Y of a topological space is <u>connected</u> if Y is connected in the <a href="/title/subspace">subspace</a> topology inherited from X.
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This definition sounds a bit odd at first, but it does give the desired effect - that is, the spaces you would expected to be connected are, and vice versa.…separation axiom (idea)http://m.everything2.com/user/Evandar/writeups/separation+axiomEvandarhttp://m.everything2.com/user/Evandar2001-07-09T01:10:53Z2001-07-09T01:10:53ZSome extra comments on separation <a href="/title/axiom">axiom</a>s:
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<u>T<sub>1</sub></u> Another way of phrasing the T<sub>1</sub> axiom is: for each x in X, {x} is <a href="/title/topological+space">closed</a>. It's not hard to show this is equaivalent to the T<sub>1</sub> axiom as given above, and it's often much easier to work with and to <a href="/title/visualize">visualize</a>.
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A simpler example of a space which is T<sub>1</sub> but not T<sub>2</sub> is the set <b>R</b> of <a href="/title/real+numbers">real numbers</a> with the topology where the <a href="/title/topological+space">open sets</a> are the empty set and all subets U of <b>R</b> such that <b>R</b>\U is finite. It's clear that singleton sets are closed, so this is T<sub>1</sub> by the comment above, but it's not T<sub>2</sub> because any two nonempty open sets must have <a href="/title/infinite">infinitely many</a> real numbers in common.
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<u>T<sub>2</sub></u> This axiom, the <a href="/title/Hausdorff">Hausdorff</a> axiom, is the minimal condition under which <a href="/title/sequence">sequence</a>s (or <a href="/title/net">net</a>s) <a href="/title/converge">converge</a> to at most one point. In spaces that are not T<sub>2</sub>, very strange things can happen…Luring lottery (idea)http://m.everything2.com/user/Evandar/writeups/Luring+lotteryEvandarhttp://m.everything2.com/user/Evandar2001-04-15T02:21:55Z2001-04-15T02:21:55ZIt's a <a href="/title/lottery">lottery</a> with a prize fund of $1 000 000. You can enter as many times as you want. There's only one <a href="/title/catch">catch</a>: the amount the winner gets is the the <a href="/title/a+million+dollars">million dollars</a>, divided by the total number of entries.
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This was a game introduced by <a href="/title/Douglas+Hofstadter">Douglas Hofstadter</a> in <a href="/title/Scientific+American">Scientific American</a> in August of 1983, and described in <a href="/title/Martin+Gardner">Martin Gardner</a>'s <a href="/title/Mathematical+Games">Mathematical Games</a> column. Scientific American put up the money, and anyone could enter; all you had to was send them a postcard containing the number of entries you would like to make.
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When someone sends in a postcard with a number on it, they get that number of tickets with their name on them put into a <a href="/title/hat">hat</a>. When all the entries are in, one <a href="/title/ticket">ticket</a> is removed from the hat at random, and the person whose name is on that ticket wins the value of the pot, which is $1 000 000 divided by the total number of tickets in the hat.
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Simple as that.
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Of course, it's not as simple as that. The more entries are sent in, the less…Axiom of Separation (idea)http://m.everything2.com/user/Evandar/writeups/Axiom+of+SeparationEvandarhttp://m.everything2.com/user/Evandar2001-04-14T18:20:38Z2001-04-14T18:20:38ZOne of the <a href="/title/zf">Zermelo-Frankel</a> <a href="/title/axiom">axiom</a>s of <a href="/title/set+theory">set theory</a>, also known as the subset axiom. Like the <a href="/title/axiom+of+replacement">axiom of replacement</a>, the axiom of separation is actually an <a href="/title/infinite">infinite</a> set of axioms, an '<a href="/title/axiom+schema">axiom schema</a>'. Specifically, for every formula P(x) of <a href="/title/the+language+of+set+theory">the language of set theory</a>, the following is an axiom:
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<b><a href="/title/Axiom">Axiom</a></b> If A is a set, then there is a set B such that y is in B if and only if y is in A and P(y) is true.
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The set B is normally denoted {y in A: P(y)}.
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In other words, given a set A, we can separate out the elements of A satisfying a property P, and put them in a new set.
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Notice that we have to pick the elements from some set A. In general, we can't take the collection of <i>all</i> sets satisfying some property, for if we could then we could take <a href="/title/russell%2527s+paradox">the set of all sets not members of themselves</a>, giving <a href="/title/Russell%2527s+paradox">Russell's paradox</a>. The axiom of separation as I've given it above was formulated to avoid this problem.Axiom of Foundation (idea)http://m.everything2.com/user/Evandar/writeups/Axiom+of+FoundationEvandarhttp://m.everything2.com/user/Evandar2001-04-14T15:18:57Z2001-04-14T15:18:57ZOne of the <a href="/title/zf">Zermelo-Frankel</a> <a href="/title/axiom">axiom</a>s of <a href="/title/set+theory">set theory</a>. Also know as the axiom of regularity, the axiom of foundation is not needed in general <a href="/title/mathematics">mathematics</a> but affords a certain peace of mind to set theorists.
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<b><a href="/title/Axiom">Axiom</a></b> Whenever A is a <a href="/title/nonempty">nonempty</a> set, there is a set y in A such that the <a href="/title/intersection">intersection</a> of y and A is empty.
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Pick any set. If it's not empty, you can choose any of its members, and if that set isn't empty you can choose one of <i>its </i>members, and so on. From the axiom of foundation it can be shown that this process must eventually (ie. after finitely many steps) finish, when you have to pick the empty set.
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Consequently, you can't have any set being a member of itself, or sets a and b such that a is in b and b is in a, or any nonsense like that.
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Without the axiom of foundation, it's consistent that there is a set a such that a={a}. You can't actually <a href="/title/v%253Dl">construct</a> such a set, but you can't prove it's not there either.set (idea)http://m.everything2.com/user/Evandar/writeups/setEvandarhttp://m.everything2.com/user/Evandar2001-04-13T23:58:26Z2001-04-13T23:58:26Z<big><big><b>Set: An Introduction To The Idea Of <a href="/title/Geometry">Geometry</a> In <a href="/title/Four+Dimensions">Four Dimensions</a></b></big></big>
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Believe it or not, the simple card game of set is actually based on <a href="/title/four+dimensional">four dimensional</a> <a href="/title/geometry">geometry</a>. No really. And you thought it was just a <a href="/title/waste+of+time">waste of time</a>...
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For those of you who haven't played it, set is played with 81 cards, as described by <a href="/title/%252Fdev%252Fjoe">/dev/joe</a> above. Now 81 = 3<sup>4</sup> = 3x3x3x3, which is no coincidence - but you'll see why this is later, if you haven't already.
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To play, you lay 12 cards out on the table, and the players have to find a set, which is a collection of three cards from the 12, where for each <a href="/title/category">category</a> (colour, number, shape or pattern), <b>either</b> they are all the same (for example, all three are red) <b>or</b> they are all different (one is diamonds, one ovals and one squiggles). And to make a set, you have to do this for all categories at once.
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So the triplets that aren't sets are the ones where two are red and one is blue, or two are half-shaded and…