[自問自答] K-topology

我剛剛聽到有人在討論 K theory
突然靈光一閃 回想起很久以前念到的 K topology
想說 不會這兩個東西可以經由奇怪的關連聯在一起吧
(可是這個定義感覺沒有什麼關係啊...)

剛剛去查 wiki
呃 好像發現他們真的沒有關係
只是拓樸學家用來構造反例的產物

如:

3. (R, T) is Hausdorff but not regular.
    表示 T2 < T3

4. Surprisingly enough, (R, T) is a connected topological space.
   However, (R, T) is not path connected
    表示 雖然 p.conn. => conn. 但反過來不對

5. Note also that (R, T) is not locally path connected.
   It is also not locally connected at {0},
   but it is locally connected everywhere else

6. The closed interval [0,1] is not compact as a subspace of (R, T)
   since it is not even limit point compact

7. In fact, no subspace of (R, T) containing K can be compact.
   If A were a subspace of (R, T) containing K,
      K would have no limit point in A
   so that A can not be limit point compact.
   Therefore, A cannot be compact

8. The quotient space of (R, T) obtained by collapsing K to a point
   is not Hausdorff.

9. 此外 (R, T) 還是 locally metrizable, 即使他不 metrizable.