T1 - A topological space is said to be T1 if for any points x, y in the space, there exists an open set U such that x is contained in U but y is not contained in U.
Weakly Countably Compact - A topological space is said to be Weakly Countably Compact if every infinite subset of the space has a limit point. *note* These are not my definitions, but I am taking this to mean that the set merely HAS a limit point, not that it necessarily CONTAINS said limit point.
Countably Compact - A topological space is said to be Countably Compact if every countable open cover admits a finite subcover.
The problem I am trying to tackle here is that every T1, Weakly Countably Compact space is necessarily Countably Compact. The difficulty thus far has been that the T1 condition is an existence condition; that is, that such an open set must exist, not that it must be present in some given cover. I think I am very near a solution, though.
I suppose for those of you not versed in topology, a bit more might help. A topological space is merely a set, say, X, equipped with a topology. What is a topology? It is simply a subset, T, of the power set of X, P(X) such that these three conditions hold:
1) T contains both X and the empty set
2) T is closed under finite intersections
3) T is closed under arbitrary unions
If a set is an element of the topology, that set is said to be open. The complement of an open set is a closed set and vice versa. Theoretically, this should be enough for you to get me a proof, which I will expect soon.
But, for a familiar example, take the set of real numbers (which you can visualize as a line if you like) as X and let T={U|if x is an element of U, there exists an interval (a,b) such that x is an element of (a,b) and (a,b) is a subset of U}. This is actually called the standard topology on R, and is really the origin of all topology. You can confirm this is indeed a topology, which is a worthwhile exercise.
Anyway, the problem I've laid out here isn't going to be so easy to draw as the real line, as it turns out that the real line under the standard topology isn't weakly countably compact. You can confirm this, too, if you'd like. It is, however, T1, which would be good to think about, I suppose. Since I can't draw a nice picture of our situation, have instead a picture of curry udon that I made tonight:
Now, for a problem of a different sort, for those of you not inclined to mathematical brain teasers. Here's a bit of a song I've written, or at least written the words to, as I've forgotten the melody I originally had in mind:
The girl was a tall Australian one
Too tall for her frame, if you know what I mean
Spoke like a movie with incomprehensible accents
I couldn't follow the plot, except for her scene
She had crimson hair just like a convict
Her father was a resident of the penetentiary
She was driven to sadness, and from sadness to madness and drinking
And I was driven to her by a car when she drove into me
Well, the sun shone down on that Australian outback
The seasons there are all in reverse
I was thinking of her sitting at the hospital piano
She was thinking of me, but I was thinking of her first
The girl was a tall Australian one
With features too sharp to make her aboriginee
She was probably wearing her denim dress when we parted
She always did for some reason when meeting me
She freckles easily and so she wears a lot of sunscreen
And it matches with her crimson hair so hot
If she walked through the desert she'd look like she's on fire
With the air melting around her, even though it's not.
It's not a particularly good song, I'll admit, and the syllables may seem too plentiful, but keep in mind I write for how I sing and I sing like Dylan in a funny kind of way. The problem here is what should happen next. I'm not really sure. I'm not asking for you to finish the song for me, just for an idea. I can deal with the actual construction my self. I'm also not asking for critique, but feel free to leave that if you want. Here's an unrelated picture of some banana juice that I bought the other day:
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5 comments:
No offense, but I couldn't get through the first paragraph of this entry....but, I'm still REALLY interested in what you are doing!
Problem 1: I think you are the only person I know who would right a math problem like this on a blog that has nothing inherently to do with mathematics. Thus, to you I say, well played, sir.
Problem 2: Well, you have started the song with what amounts to a descriptive background, so as a follower of Dylan, perhaps you now need to give this Australian beauty a story in forward motion. Perhaps a storyline about the relationship's beginning, middle and end? Just a thought.
Marisa, I'm sorry if it was a bit hard to get through. I tried to lay it out so that anyone could follow it if they took the time, but I understand people not really having an interest.
Il2va, thanks. The grammar nazi in me would be remiss, though, if he didn't point out that it should be "write a math problem." The usage nazi in me, though, would like to point out that that is really a usage problem less than a grammar one, since one could indeed say the sentence makes sense if we accept that there was something inherently wrong in the problem which needed righting.
Furthermore, il2va, thanks for the advice on the song. Since the song is about someone who is a bit crazy, and the meeting of the characters comes when she hits the narrator with a car, I was thinking of having the end of the story involve her being put away or something of the like.
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