An Introduction to K-theory
... A well-known theorem of Minkowski asserts that Cl(K) is finite for any number field K (cf. [5]). Computing class groups is devilishly difficult. We do know that there only finitely many cyclotomic fields (i.e., of the form Q(ζn ) obtained by adjoining a primitive n-th root of unity to Q) with class ...
... A well-known theorem of Minkowski asserts that Cl(K) is finite for any number field K (cf. [5]). Computing class groups is devilishly difficult. We do know that there only finitely many cyclotomic fields (i.e., of the form Q(ζn ) obtained by adjoining a primitive n-th root of unity to Q) with class ...
Lifting of maps in topological spaces
... 7. Not all maps can be lifted Just after defining the notion of Lifting Of Maps, I had mentioned that not all maps can be lifted. I feel that it is very exciting to see an actual example of a situation where it can fail. One must note that in almost any example that can be given for a situation whe ...
... 7. Not all maps can be lifted Just after defining the notion of Lifting Of Maps, I had mentioned that not all maps can be lifted. I feel that it is very exciting to see an actual example of a situation where it can fail. One must note that in almost any example that can be given for a situation whe ...
preopen sets and resolvable spaces
... (i) (X, τ ) contains an open, dense and hereditarily irresolvable subspace. (ii) Every open ultrafilter on X is a base for an ultrafilter on X . (iii) Every nonempty open set is irresolvable. (iv) For each dense subset D of (X, τ ) , int D is dense. (v) For every A ⊆ X , if int A = ∅ then A is nowhe ...
... (i) (X, τ ) contains an open, dense and hereditarily irresolvable subspace. (ii) Every open ultrafilter on X is a base for an ultrafilter on X . (iii) Every nonempty open set is irresolvable. (iv) For each dense subset D of (X, τ ) , int D is dense. (v) For every A ⊆ X , if int A = ∅ then A is nowhe ...
Submaximality, Extremal Disconnectedness and Generalized
... α-clS = S Scl(int(clS)) and! ! sclS = S int(clS), pclS = S cl(intS) and spclS = S int(cl(intS)). The α-interior T of S ⊆ X is the largest α-open set contained in S, and we have α-intS = S int(cl(intS)). It is worth mentioning that the collection of all α-open subsets of (X, τ ) is a topology τ α on ...
... α-clS = S Scl(int(clS)) and! ! sclS = S int(clS), pclS = S cl(intS) and spclS = S int(cl(intS)). The α-interior T of S ⊆ X is the largest α-open set contained in S, and we have α-intS = S int(cl(intS)). It is worth mentioning that the collection of all α-open subsets of (X, τ ) is a topology τ α on ...
16. Maps between manifolds Definition 16.1. Let f : X −→ Y be a
... Theorem 16.9. Let f : X −→ Y be a local homeomorphism of manifolds. Then f has the lifting property if and only if f is an unramified cover. We have already shown one direction of (16.9). To prove the other direction we need the following basic result: Theorem 16.10. Let f : X −→ Y be a local homeo ...
... Theorem 16.9. Let f : X −→ Y be a local homeomorphism of manifolds. Then f has the lifting property if and only if f is an unramified cover. We have already shown one direction of (16.9). To prove the other direction we need the following basic result: Theorem 16.10. Let f : X −→ Y be a local homeo ...
Topological vector spaces - SISSA People Personal Home Pages
... Proof. If X has a neighborhood V which is compact, then by the finite covering property V ⊂ (x1 + V /2) ∪ · · · ∪ (xn + V /2). Let Y be the vector space generated by (x1 , . . . , xn ). The previous formula reads as V ⊂ Y + V /2. Moreover Y is closed. By linearity, we have also V /2 ⊂ Y + V /4, so t ...
... Proof. If X has a neighborhood V which is compact, then by the finite covering property V ⊂ (x1 + V /2) ∪ · · · ∪ (xn + V /2). Let Y be the vector space generated by (x1 , . . . , xn ). The previous formula reads as V ⊂ Y + V /2. Moreover Y is closed. By linearity, we have also V /2 ⊂ Y + V /4, so t ...
