Covering Space

Then there is a bijection between the set of basepoint preserving isomorphism classes of path connected covering spaces p.

Covering space. X e x satisfying the following condition. If the mapping q. X x 0 and the set of subgroups of ˇ 1 x x.

For a covering space the inverse image of some open set in the base b needs to be by the definition a disjoint union. X y such that any point in y has a neighborhood n whose preimage in x is a collection of disjoint sets which are homeomorphic to n. Dec 19 2010 7.

D x is a universal cover of the space x and the mapping p. Every covering space even in the more general sense not requiring any connectedness axiom is an etale space but not vice versa. So why is this helpful.

That gives sort of an infinite spiral over the circle. Theorem classification of covers. A covering space x of a space y is a space with a map p.

To every subgroup of 1 b b there is a covering space of b so that the induced subgroup is the given one. So the universal cover of a circle is given by the set of all paths on a circle starting at 1 mod homotopy. A simply connected covering space if and only if y is semilocally simply connected.

Let x be path connected locally path connected and semilocally simply connected. If we did not pin down a basepoint in a a covering space would only give a well defined conjugacy class of subgroup of 1 b b. Even if the stalks of the etale space are.