## What are path-connected space?

A path-connected space is a stronger notion of connectedness, requiring the structure of a path. A path from a point x to a point y in a topological space X is a continuous function ƒ from the unit interval [0,1] to X with ƒ(0) = x and ƒ(1) = y.

## Is the quotient map Injective?

Quotient maps could be not injective. However, when you have a quotient map f:X→Y then f induces a homeomorphism between the quotient space X/∼f and Y, where x1∼fx2 iff f(x1)=f(x2).

**What is quotient topology explain with example?**

Let q:X→X/∼ be the quotient map sending a point x to its equivalence class [x]; the quotient topology is defined to be the most refined topology on X/∼ (i.e. the one with the largest number of open sets) for which q is continuous. …

### Are quotient maps continuous?

In particular, quotient maps are continuous. Moreover, if g : X → Y is a quotient map, the topology on Y is completely determined by the function g and the topology on X.

### How do you prove a path is connected?

(8.08) We can use the fact that [0,1] is connected to prove that lots of other spaces are connected: A space X is path-connected if for all points x,y∈X there exists a path from x to y, that is a continuous map γ:[0,1]→X such that γ(0)=x and γ(1)=y.

**Is the closure of a path-connected set path-connected?**

The closure of a path-connected subset Y of X is not necessarily path-connected. Thankfully, the next result does carry over. is a continuous map of topological spaces and X is path-connected, then so is f(X).

## Is the quotient map an open map?

The map f : X → Y is a closed map if for each closed set A ⊆ X the set f(A) is closed in Y . Note. If p : X → Y is continuous and surjective and p is either open or closed, then p is a quotient map. However, there are quotient maps that are neither open nor closed (see Munkres Exercise 22.3).

## Are quotient maps closed?

A map f:X→Yis called proper, iff preimages of compact sets are compact. It is called quotient map, iff a subset V⊂Y is open, if and only if its preimage f−1(V) is open. And it is called closed, iff it maps closed sets to closed sets.

**What is meant by quotient set?**

A quotient set is a set derived from another by an equivalence relation. Let be a set, and let be an equivalence relation. The set of equivalence classes of with respect to is called the quotient of by , and is denoted . A subset of is said to be saturated with respect to if for all , and imply .

### Is the quotient space compact?

If X is compact (connected), then the quotient space X/∼ is also compact (connected). The projection map p : X → X/∼, is continuous and onto and the continuous image of a compact (connected) space is compact (connected).

### What is connected and path connected?

All the points in a path connected component can be connected to each other. A set, or space, is path connected if it consists of one path connected component. The continuous image of a path is another path; just compose the functions. The image of a path connected component is another path connected component.

**Is the empty set path connected?**

Conventionally the empty set is path-connected. if x > 0, f(x)=0 if x ≤ 0.

## How is a quotient space related to an identification space?

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or “gluing together” certain points of a given topological space. The points to be identified are specified by an equivalence relation.

## Can a quotient space be neither open or closed?

It is easy to construct examples of quotient maps that are neither open nor closed. For topological groups, the quotient map is open. In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of X need not be inherited by X /~, and X /~ may have separation properties not shared by X.

**How are continuous maps defined in a quotient space?**

The continuous maps defined on X /~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is copiously used when studying quotient spaces.

### Which is an example of quotient space in topology?

A generalization of the previous example is the following: Suppose a topological group G acts continuously on a space X. One can form an equivalence relation on X by saying points are equivalent if and only if they lie in the same orbit. The quotient space under this relation is called the orbit space, denoted X / G.