What do you mean by covering spaces?

Given a topological space X, we’re interested in spaces which “cover” X in a nice way. Roughly speaking, a space Y is called a covering space of X if Y maps onto X in a locally homeomorphic way, so that the pre-image of every point in X has the same cardinality.

What is fundamental group of covering spaces?

The fundamental group is one of the most important topological invariants of a space, and a rather accessible one at that. It is essentially a “group of loops,” consisting of all possible loops in a space up to homotopy. Definition 2.1. A loop (sometimes called a closed path) in X is a path f with f(0) = f(1).

Why do we cover space when studying?

One answer is that it provides a wealth of group actions, which will allow us to study the structure of groups by understanding properties of the actions that they have on spaces.

What is covering projection explain it with example?

A map from a space to a set is locally constant if it is continuous when the set is given the discrete topology. A covering projection is a map which is, locally in the base, a trivial covering projection. Examples. e : R → S1 by t ↦→ e2πit. pn : S1 → S1 by z ↦→ zn.

Is covering map unique?

Universal covers The map f is unique in the following sense: if we fix a point x in the space X and a point d in the space D with q(d) = x and a point c in the space C with p(c) = x, then there exists a unique covering map f : D → C such that p ∘ f = q and f(d) = c.

Is universal cover unique?

Is a covering map continuous?

Definition 9. A covering map of topological spaces p : ˜X −→ X is a continuous map such that there exists an open cover X = ⋃α Uα such that p−1(Uα) is a disjoint union of open sets (called sheets), each homeomorphic via p with Uα.

Is covering map open?

If is a covering space of then is an open map. Recall that a map between topological spaces is said to be open if the image of any open set in the domain is an open set in the range.

Is the identity map a covering map?

Let X be any space; let i : X X be the identity map. Then i is a covering map (of the most trivial sort). More generally, let E be the space X X {1,…,n} consisting of n disjoint copies of X. The map p : E → X given by p(x,i) = x for all i is again a (rather trivial) covering map.

Is a covering map open?

But covering maps are open, so is open. Hence, is open iff is; that is, has the quotient topology. (This argument works for any continuous map that is open and surjective.)

Are covering maps surjective?

The way covering map has been defined allows it not to be surjective (the condition holds vacuously for points with empty pre-image); the usual definition has a covering map being surjective.

Is a covering map a quotient map?

Covering maps are always quotient maps. Let be a covering, and is such that is open. But covering maps are open, so is open.