UI MATH 5400 - Quotient Spaces and Quotient Maps

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22M:132Fall 07J. SimonComments onQuotient Spaces and Quotient MapsThere are many situations in topology where we build a topological space bystarting with some (often simpler) space[s] and doing some kind of “ gluing” or“identifications”. The situations may look different at first, but really they areinstances of the same g eneral construction. In the first section below, we give someexamples, without any explanation of the theoretical/technial issues. In the nextsection, we give the general definition of a quotient spa ce and examples of severalkinds of constructions that are all special instances of this general one.1. Examples of building topological spaces with interesting shapesby starting with simpler spaces and doing some kind of gluing oridentifications.Example 0.1. I dentify the two endpoints of a line segment to form a c i rcle .Example 0.2. I dentify two opposite edges of a rectangle (i.e. a rectangular strip ofpaper) to form a cylinder.cJ. Simon, all rights reserved page 1Example 0.3. Now ide ntify the top and bottom circles of the cylinder to eac h other,resulting in a 2-dim ensional surface called a torus.Example 0.4. Start with a round 2-dimensional disk; identify the whole boundarycircle to a single point. The result is a s urface, the 2-sphere.Example 0.5. ?? Take two different spheres (in particular, disjoint from eachother); pick one point on each sphere and glue the two spheres together byidentifying the chosen point from each.cJ. Simon, all rights reserved page 2Example 0.6. Th e set of all homeomorphisms from a space X onto itself forms agroup: the operation is composition (The composition of two homeomorphisms is ahomeomorphism; functional composition is associative; the identity map I : X → Xis the identity element in the group of homeomorphisms.) The group of allself-homeomorphisms of X may have interesting subgroups. When we specify s ome[sub]group of homeomorphisms of X that is isomorphic to s ome abstract group G,we call this an action of the group G on X. Note that in this situation, we areviewing the elements of G as homeomorphisms of X, and the group operation ◦ in Gas function composi tion ; so, in particular, f or each g, h ∈ G, x ∈ X, we are insistingthat (g ◦ h)(x) = g(h(x)).When we have a group G acting on a space X, there is a “natural” quotient space.For each x ∈ X, let Gx = {g(x) | g ∈ G}. View each of these “orbit” sets as a singlepoint in s ome new space X∗.2. Definition of quotient spaceSuppose X is a topological space, and suppose we have some equivalence relation“∼” defined on X. Let X∗be the set of equivalence classes. We want to define aspecial topology on X∗ , called t he quotient topology. To do this, it is convenient tointroduce the functionπ : X → X∗defined byπ(x) = [x] ,that is, π(x) = the equivalence class containing x.This can be confusing, so say itover to yourself a few times: π is a f unction from X into the power set of X; itassigns to each point x ∈ X a certain subset of X, namely the equivalence classcontaining the point x. Since each x ∈ X is contained in exactly one equivalenceclass, the f unction x → [x] is well-defined. At the risk of belaboring the obvious,since each equivalence class has at least one member, the function π is surjective.Before talking about the quotient topology, let’s look at several examples of thequotient sets X∗.Example 0.7. Let X = {a, b, c}, a set with 3 points. Partition X into twoequivalence classes: {1, 3 } and {2}. So X∗has two elements, call them O and T.The function π isπ(1) = O, π(2) = T, π(3) = O .Example 0.8. In our previous example 0.1, one equival e nce clas s has two elements;every other equivalence class is a singleton. Likewise, in example ??, oneequivalence class has two elements and all the others are singletons.Example 0.9. I n our previous example 0.2, each equivalence class coming frompoints on the vertical edge s being identified con sists of two points; all otherequivalence classes are singletons.cJ. Simon, all rights reserved page 3Example 0.10 (shrinking a set to a point). Let X be any space , and A ⊆ X. Thethe quotient space X/A is the set of equivalen ce classes [x], where [x] = A if x ∈ Aand [x] = {x} if x /∈ A. The set X∗has one “giant” point A and the rest are justthe points of X − A. Th is is the situation in example 0.4Example 0.11. Let X be the real line R1. Let G be the ad ditive group of integers,Z. Define an action of G on R b y n(x) = x + n for each n ∈ Z, x ∈ R. Here the setGx consists of all integer translates of the point x. Note that the sets Gx do form apartition of X. That is, the relation x ∼ y ⇐⇒ there exists g ∈ G with g(x) = y isan equivalence relation on X. [ Unas signed exercise: Check this claim; it depends onthe fact that G is a group.]However, unlike our previo us examples, it may be not so obvious what a geometric“picture” of X∗looks like: the number of poi nts is the same as the half-open interval[0, 1); but what should the topology be??3. The quotient topologyIf we think of constructing X∗by actually picking up a set X and squishing someparts together, we would like t he passage fro m X to X∗to be continuous. We makethis precise by insisting that the projection mapπ : X → X∗π(x) = [x]be continuous. This puts an obligation on the topology we assign to X∗: If a set Uis open in X∗then π−1(U) is open in X. (Think of this as an “upper bound” onwhich sets in X∗can be open.) We define the quotient topology on X∗by letting allsets U that pass this test be admitted.A set U is open in X∗if and only if π−1(U) is open in X. The quotient topology onX∗is the finest topology on X∗for which the projection map π is continuous.We now have an unambiguously defined special topology on the set X∗ofequivalence classes. But that does not mean that it is easy to recognize whichtopology is the “right” one. Going back to our example 0.6 , the set of equivalenceclasses (i.e. orbit sets Gx) is in 1-1 correspondence with the points of the half-openinterval [0, 1). But t hat does not imply that the quotient space, with the quotienttopology, is homeomorphic to the usual [0, 1). To understand how to recognize thequotient spaces, we introduce the idea of quotient map and then develop the text’sTheorem 22.2. This theorem may look cryptic, but it is the tool we use to provethat when we think we know what a quotient space


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UI MATH 5400 - Quotient Spaces and Quotient Maps

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