Covering Space

A covering space is a universal covering space if it is simply connected.

Covering space. The name universal cover comes from the following important property. Let x be path connected locally path connected and semilocally simply connected. So the universal cover of a circle is given by the set of all paths on a circle starting at 1 mod homotopy. Every point x x has an open neighborhood u x x such that p 1 u x is a disjoint union of open sets each of which is mapped by p.

That gives sort of an infinite spiral over the circle. D x is a universal cover of the space x and the mapping p. For a covering space the inverse image of some open set in the base b needs to be by the definition a disjoint union. Then there is a bijection between the set of basepoint preserving isomorphism classes of path connected covering spaces p.

If the mapping q. A covering space or cover of a space x is a space xe together with a map p. A simply connected covering space if and only if y is semilocally simply connected. Covering spaces are constructed by taking paths in the base space.

Even if the stalks of the etale space are. A covering space x of a space y is a space with a map p. Every covering space even in the more general sense not requiring any connectedness axiom is an etale space but not vice versa. To every subgroup of 1 b b there is a covering space of b so that the induced subgroup is the given one.

D c such that p f q. C x is any cover of the space x where the covering space c is connected then there exists a covering map f. 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. Dec 19 2010 7.

X e x satisfying the following condition. Well in our example we can say that the real line covers the circle from the pink pictures. 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.