What do you mean by covering space?

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. It turns out that the covering spaces of X have a lot to do with the fundamental group of X.

What is covering map in mathematics?

In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function from a topological space to a topological space such that each point in has an open neighborhood evenly covered by (as shown in the image).

What is covering projection explain it with example?

Definition. A covering projection is a continuous map p : E → B with the. property that B admits an open cover U such that for each U ∈ U there is a. set F and a locally constant map q : p-1U → F such that.

What is fundamental group of covering space?

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).

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.

What does double cover mean?

In gridiron football double coverage is a state of defensive playcalling wherein two defensive players are assigned to “cover” one offensive player. This situation is often seen with standout wide receivers and running backs. It is extremely rare to have two defensive backs man-cover a single receiver.

Is a covering map surjective?

Is a homeomorphism a covering map?

Also, a proper local homeomorphism from a Hausdorff space onto a locally path connected Hausdorff space is a covering map.

Is a covering map Surjective?

Is a Homeomorphism a covering map?

Is 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. Hence, is open iff is; that is, has the quotient topology.

What does double cover mean in football?

defensive playcalling
In gridiron football double coverage is a state of defensive playcalling wherein two defensive players are assigned to “cover” one offensive player. This situation is often seen with standout wide receivers and running backs.

How to prove the theory of covering spaces?

In §10 we prove that every filtered group is isomorphic to the Poincaré filtered group of some regular covering space of a connected topo- logical space [Theorem 2]. We use this result to translate topological ques- tions on covering spaces into purely algebraic questions on filtered groups.

When is a topological space called a regular space?

Regular space. In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C can be separated by neighborhoods.

Which is the best definition of a covering map?

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function from a topological space to a topological space such that each point in has an open neighbourhood evenly covered by (as shown in the image); the precise definition is given below.

Which is an important application of covering space?

Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if .

What do you mean by covering space? 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. It turns out that the covering spaces of X have a lot to…