European beer isn't that much better than American.
That's a fairly generalizing statement. While it is true that most e.g. Danish (Carlsberg) and Dutch (Heineken) beer is only marginally more interesting than the typical US light lager, there is a much broader spectrum of mainstream beers from countries such as Ireland (ale, stout), Germany (wheat, black, Oktoberfest) and Belgium (just about anything you can think of).
If you want something light for a hot summer day, try a nice plain ale, a belgian wit, a gueuze (on the other hand don't try a gueuze unless you know what you get yourself into)... there are plenty of much better beer very well suited for that purpose.
Why doesn't Microsoft just tell people to switch to Ubuntu and use Firefox?
You wouldn't want that to happen. If everybody used Ubuntu and Firefox, malware creators would start finding the exploits and workarounds there, and suddenly you (assuming, perhaps incorrectly your post reflects your own setup) would be in harm's way as well. Microsoft probably has shoddy security and such, but the main reason their products are being attacked is their dominance.
I must admit I didn't know of any way to speak of convergence without the notion of a metric. How is that possible?
In a topological space you have a notion of a neighbourhood of a point, i.e. a set containing the point in it's interior. You then say that the sequence (x_n) converges to the point x if for every neighbourhood U of x, there is a number N, such that x_n is in U whenever n>N. Basically this is a translation of the epsilon-N-formalism for convergence in metric spaces (since in a metric space U is a neighbourhood of x exactly when there is an epsilon such that the epsilon-ball around x is contained in U).
More general than what?
More general than 1st-countable spaces. That is general topological spaces with no structure but the existence of a family of open sets.
And do you mean we need the "net" to replace the sequence?
Yes. The concept of a convergent sequence is not in general strong enough to capture topological properties such as continuity (i.e. the preimage of an open set is open), closedness (the complement is open), and compactness (every open cover has a finite subcover), but replacing sequences by nets, you actually get the classical characterisations of these properties known from metric spaces; a function is continuous iff x_i->x implies f(x_i)->f(x), A is closed iff any net (x_i) in A, which converges in your larger space has the limit contained in A, and C is compact iff any net has a convergent subnet with limit in C. Nets are really nice objects for doing point set topology.
However, one must still be able to define a sequence (a function from "the set of all natural numbers" to "the topological space in question"), since it doesn't really require much of the space, right?
Yes, no problems there. They just don't provide enough information once we go to the more obscure topological spaces.
Your version is a more general one. In a metric space, where your parent's version is applicable, the two are equivalent.
Math geek warning!
Actually the equivalence goes much further than for metric spaces. In all topological spaces you have a sense of convergence of a sequence, and so it makes sense to ask the question "Does x_n->x imply f(x_n)->f(x)?". If f is continuous, the answer is always yes, but the converse need not be true in general - it is however true if the topological space is a so-called 1st-countable space, e.g. if the space is metric (as you said).
For even more general topological spaces you need the concept of a net, but in that setting the equivalence is actually total; a function f is continuous if and only if x_i->x implies f(x_i)->f(x) for every net (x_i).
This is sometimes mis-stated as 'you can draw the graph without taking your pen off the paper'.
That's not a mis-statement in the case of a real function of a real variable. It's not that informative, but definitely correct in the sense that a function (real etc.) is continuous iff the graph is path-connected (i.e. every two points on the graph can be connected by a continuous path (and by saying 'continuous path' I have of course made the definition self-referential and thus silly, but it is still true)).
Who would dare to predict what a "computer" will look like in 10 years time?
I predict that they will weigh no more than 1.5 tons, and modern transistors (perhaps even smaller than today!) will have replaced nearly all the radio tubes!
If that was what I meant, that would have been what I said. The caricatures definitely was offensive, and the Danish PM did handle the situation very arrogantly initially by refusing to see the ambassadors, but seing as this is nowhere near the first - or the worst - instance of Mohammad drawings, the reaction is absolutely out of proportions, and I do believe it has more to do with middle eastern interior politics than with Denmark as such.
Well, it has never been about the caricatures (well, it has, but not really). The muslim world - or at least part of it, centered around the more fundamentalist governments - need an outside enemy. Egypt really isn't outside, and the Saudis don't want to anger the US - enter Denmark. A relatively small country with basically no impact whatsoever on life in Saudi Arabia; the perfect target.
Badly formatted post. Sorry about that - we try again.
how many times was Firefly preempted by football during its first run?
Not that many, really - Firefly was on friday night. I think you're thinking of Futurama (yet another great show killed by Fox).
With how poorly Fox treated Firefly (and Angel, well, how they treated Joss in general those last couple years),
Angel was on The WB. Joss was only mistreated by Fox in that one instance. Tim Minear (co-creator of Firefly) however had Firefly, Wonderfalls (great show - get the DVD box if you have the chance) and The Inside killed of by Fox during their first seasons. Now there's a guy being mistreated.
how many times was Firefly preempted by football during its first run?
Not that many, really - Firefly was on friday night. I think you're thinking of Futurama (yet another great show killed by Fox).
With how poorly Fox treated Firefly (and Angel, well, how they treated Joss in general those last couple years),
Angel was on The WB. Joss was only mistreated by Fox in that one instance. Tim Minear (co-creator of Firefly) however had Firefly, Wonderfalls (great show - get the DVD box if you have the chance) and The Inside killed of by Fox during their first seasons. Now there's a guy being mistreated.
Slashdot Mirror
If you want something light for a hot summer day, try a nice plain ale, a belgian wit, a gueuze (on the other hand don't try a gueuze unless you know what you get yourself into)... there are plenty of much better beer very well suited for that purpose.
I must admit I didn't know of any way to speak of convergence without the notion of a metric. How is that possible?
In a topological space you have a notion of a neighbourhood of a point, i.e. a set containing the point in it's interior. You then say that the sequence (x_n) converges to the point x if for every neighbourhood U of x, there is a number N, such that x_n is in U whenever n>N. Basically this is a translation of the epsilon-N-formalism for convergence in metric spaces (since in a metric space U is a neighbourhood of x exactly when there is an epsilon such that the epsilon-ball around x is contained in U).
More general than what?
More general than 1st-countable spaces. That is general topological spaces with no structure but the existence of a family of open sets.
And do you mean we need the "net" to replace the sequence?
Yes. The concept of a convergent sequence is not in general strong enough to capture topological properties such as continuity (i.e. the preimage of an open set is open), closedness (the complement is open), and compactness (every open cover has a finite subcover), but replacing sequences by nets, you actually get the classical characterisations of these properties known from metric spaces; a function is continuous iff x_i->x implies f(x_i)->f(x), A is closed iff any net (x_i) in A, which converges in your larger space has the limit contained in A, and C is compact iff any net has a convergent subnet with limit in C. Nets are really nice objects for doing point set topology.
However, one must still be able to define a sequence (a function from "the set of all natural numbers" to "the topological space in question"), since it doesn't really require much of the space, right?
Yes, no problems there. They just don't provide enough information once we go to the more obscure topological spaces.
Your version is a more general one. In a metric space, where your parent's version is applicable, the two are equivalent.
Math geek warning!
Actually the equivalence goes much further than for metric spaces. In all topological spaces you have a sense of convergence of a sequence, and so it makes sense to ask the question "Does x_n->x imply f(x_n)->f(x)?". If f is continuous, the answer is always yes, but the converse need not be true in general - it is however true if the topological space is a so-called 1st-countable space, e.g. if the space is metric (as you said).
For even more general topological spaces you need the concept of a net, but in that setting the equivalence is actually total; a function f is continuous if and only if x_i->x implies f(x_i)->f(x) for every net (x_i).
This is sometimes mis-stated as 'you can draw the graph without taking your pen off the paper'.
That's not a mis-statement in the case of a real function of a real variable. It's not that informative, but definitely correct in the sense that a function (real etc.) is continuous iff the graph is path-connected (i.e. every two points on the graph can be connected by a continuous path (and by saying 'continuous path' I have of course made the definition self-referential and thus silly, but it is still true)).
Silly article summary, confusing connectedness with path-connectedness.
And just to combine stuff we could have a dense, uncountable set of measure zero as well (e.g. the union of the Cantor set and the rationals).
Who would dare to predict what a "computer" will look like in 10 years time?
I predict that they will weigh no more than 1.5 tons, and modern transistors (perhaps even smaller than today!) will have replaced nearly all the radio tubes!
You mean to say they are being a bunch of liars?
If that was what I meant, that would have been what I said. The caricatures definitely was offensive, and the Danish PM did handle the situation very arrogantly initially by refusing to see the ambassadors, but seing as this is nowhere near the first - or the worst - instance of Mohammad drawings, the reaction is absolutely out of proportions, and I do believe it has more to do with middle eastern interior politics than with Denmark as such.
Looks like there is a double-standard.
Well, it has never been about the caricatures (well, it has, but not really). The muslim world - or at least part of it, centered around the more fundamentalist governments - need an outside enemy. Egypt really isn't outside, and the Saudis don't want to anger the US - enter Denmark. A relatively small country with basically no impact whatsoever on life in Saudi Arabia; the perfect target.
Not that many, really - Firefly was on friday night. I think you're thinking of Futurama (yet another great show killed by Fox). Angel was on The WB. Joss was only mistreated by Fox in that one instance. Tim Minear (co-creator of Firefly) however had Firefly, Wonderfalls (great show - get the DVD box if you have the chance) and The Inside killed of by Fox during their first seasons. Now there's a guy being mistreated.
how many times was Firefly preempted by football during its first run? Not that many, really - Firefly was on friday night. I think you're thinking of Futurama (yet another great show killed by Fox). With how poorly Fox treated Firefly (and Angel, well, how they treated Joss in general those last couple years), Angel was on The WB. Joss was only mistreated by Fox in that one instance. Tim Minear (co-creator of Firefly) however had Firefly, Wonderfalls (great show - get the DVD box if you have the chance) and The Inside killed of by Fox during their first seasons. Now there's a guy being mistreated.