Prove Quotient Topology: Lee's Introduction to Smooth Manifolds

[Tedjn]

Homework Statement

This is from Lee's Introduction to Smooth Manifolds. Suppose π : X → Y is a quotient map. Prove that the restriction of π to any saturated open or closed subset of X is a quotient map.

Homework Equations

Lee defines a subset U of X to be saturated if U = π^-1(π(U)). π is a quotient map if it is surjective and continuous w.r.t the quotient topology defined by π.

The Attempt at a Solution

My interpretation is that I should prove that, if S is a saturated open or closed subset of X, then π|_S is a quotient map between S and π(S) = π|_S(S), both spaces being endowed with the subspace topology. That is, show that the quotient topology defined by π|_S is equivalent to the subspace topology. Is this correct?

I am imagining this approach in my mind, and I don't see how to use the hypothesis that S is either open or closed rather than any arbitrary subset. This worries me. Any advice?

 

[mighty2000]

Hi there, as a fellow scientist, I would approach this problem by first understanding the definitions and properties involved. Let's break down the problem step by step:

1. Definition of a quotient map:

A quotient map is a surjective and continuous function that maps a topological space onto a quotient space. In other words, it is a function that maps points from one space to another in a way that preserves the topological structure of the original space.

2. Definition of a saturated subset:

A subset U of X is saturated if it is equal to the inverse image of its image under the quotient map π, i.e. U = π-1(π(U)). In other words, the elements in U are mapped onto the same element in the quotient space.

3. Definition of a subspace topology:

A subspace topology is a topology on a subset S of a topological space X, where the open sets of S are defined to be the intersection of S with an open set of X.

Now, let's use these definitions to prove that the restriction of π to a saturated open or closed subset S of X is a quotient map:

1. Surjectivity:

Since π is a quotient map, it is by definition surjective. This means that for every element y in the quotient space π(S), there exists an element x in S that maps onto y under π. Therefore, π|S is also surjective.

2. Continuity:

Let U be an open subset of π(S). Since π is continuous, π^-1(U) is open in X. But since S is saturated, π^-1(U) = U ∩ S. This shows that π|S is continuous, since the inverse image of an open set in π(S) is open in S.

3. Quotient topology:

Since S is a saturated subset, the quotient topology on S is defined by the subspace topology. Therefore, the open sets in S are defined as the intersection of S with an open set in X. This is equivalent to the open sets in π(S) defined by the quotient topology. Hence, the quotient topology defined by π|S is equivalent to the subspace topology.

This proves that the restriction of π to a saturated open or closed subset S of X is a quotient map between S and π(S). I hope this helps!