From Urysohn’s lemma, we can deduce that for normal spaces, continu-ous functions to [0, 1] separate points.Proposition 5.36. In a normal space X, for any two points x, y ∈ X,x 6= y, we can find continuous f : X → [0, 1] with f (x) 6= f(y).Urysohn’s lemma makes it possible to find a function on a space thatroughly approximates a function on a subset.Lemma 5.37. Let X be normal and let A ⊂ X be closed. Suppose f : A →[−1, 1] is continuous. Then there is a continuous funct ion g : X → [−1/3, 1/3]such thatsupx∈A|f(x) − g(x)| ≤23.Proof. Let B = f−1([1/3, 1]), and let C = f−1([−1, −1/3]). By Urysohn’slemma, there is a function on X that is 1 on B and 0 on C; by rescalingthis function we get the desired function g.This lemma is key to the foll owing theorem:Theorem 5.38 (Tietze extensi on t h eor em ). A space X is T4if and o nly iffor every clos ed subset A and every continuous function f : A → R, there isa function g : X → R extending f , meaning g|A= f.Proof sketch. A rigorous proof of this theorem requires quoting some basi cresults from analysis, so we give a sketch instead. The “if” direction iseasy and is omitted. For th e “only if” direction, we compose f with ahomeomorphism R → (−1, 1) to get a bounded function. Assuming f isbounded, we approximate f by a function f1defined on X, using the lemma.The difference is presumably nonzero, so we approximate the difference byanother function f2and then f1+ f2is a function on X that restricts toa better approximation of f. Repeating this process infinitely many times,the error of the approximations goes t o zero exponentially fast, and theapproximations converge to the desired function g (this uses the fact fromanalysis that uniform limits of continuous functions are continuous).More on metric spacesSome last comments about separation and countability properties as theypertain to metric spaces. As we mentioned before, metric spaces are sepa-rable if and only if they are second countable. However, taking the balls ofrational radius around each point gives us the following:27Proposition 5.39. Every metric space is first countable.This means that we can characterize limit points and closures (and there-fore closed sets) in metric spaces in terms of sequences. Further, since metricspaces are Hausdorff, we can use seq ue nc es to uniquely specify points:Proposition 5.40. In a Hausdorff space, any sequence converges to at mostone point.The proof is an exercise using the definition. In particular, it f oll ows fromthese re mar ks that any point in a separable metric space can be specifieduniquely by a sequence of points in a fixed countable dense subset (like Qin R).6 Connectedness propertiesThe disjoint unionDefinition 6.1. The dis j oin t union of spaces (X, T ), (Y, S) is denoted X ⊔Y ; the point set is the disjoint union X ⊔Y of the pointsets, and the topologyhas the basis{U ∪ V |U ∈ T , V ∈ S},where U and V are considered as subsets of the disjoint copies of X and Yin X ⊔ Y .Proposition 6.2. Some observations:• There are canonical inclusions i1: X ֒→ X ⊔ Y and i2: Y ֒→ X ⊔ Y ;these are continuous.• The topology on X ⊔ Y is the finest one that makes the canonicalinclusions continuous.• The disjoint union X ⊔ Y is characterized by the following property:For any spa ce Z and any pair of continuous maps f : X → Z andg : Y → Z, there is a unique continuous function f ⊔ g : X ⊔ Y → Zthat with (f ⊔ g) ◦ i1= f and (f ⊔ g) ◦ i2= g.Connectedness propertiesDefinition 6.3. A separation of a space X is a pair of disjoint open nonemptysubsets U, V with U ∪ V = X.A space is connected if X has no separation. Otherwise it is dis connected.A subset S of a space X is connected if it is when considered as a subspace.28Proposition 6.4. R with the standard topology is connected.Proof. If R = U ∪ V were a separation, then without loss of generality therewould be some x ∈ R\U with U ∩ (−∞, x) 6= ∅. Then by replacing U withU ∩ (−∞, x) and V with V ∪ (U ∩ (x, ∞), we may assume that U is boundedabove. Let y be the least upper bound for U. Ce rt ai n ly y i s not in U (if so,an open i nterval around y is contained in U and y is not an uppe r boundfor U). If y ∈ V , then an open interval around y is contained in V , andV ∩ U 6= ∅. Then v /∈ U ∪ V , a contradiction.Example 6.5. Any disjoi nt union of nonempty spaces is disconnect e d.If X is a space and S , T ⊂ X are nonempty separated sets, then thesubspace topology on S ∪ T is discon n ect e d.Note that the only properties of R that we used in proving connectednesswere the least upper bound property, and the fact that an open intervalaround any point p contains points strictly less than and s t ri ct l y greaterthan p. An ordered set with these properties is called a linear continuum,and it follows that all linear continua are connected in the order topology.Proposition 6.6. If S ⊂ X is connected and T ⊂ X with S ⊂ T ⊂¯S, thenT is connected.Proof. Suppose T = U ∩ V is a separation of T . Any point in T is in Sor a limit point of S. If x ∈ U is a limit point of S, then since U is open,U ∩S 6= ∅. Similarly, V ∩S 6= ∅. So S = (U ∩S)∪(V ∩S) is a separation .Proposition 6.7. The image of a connected space under a continuous func-tion is connected.Proof. Suppose f : X → Y is a continuous function. If f(X) is disconnectedwith a s ep arat i on f(X) = U ∪V , then X = f−1(U)∪f−1(V ) is a separation.Example 6.8. Since any open interval is homeomorphic to R, any openinterval is connected. Since any i nterval is a subset of the closure of anopen i nterval, any interval is connected. Alternatively, we can s ee that anyinterval is connected bec aus e any interval is a continuous image of R.Since intervals are connected, we can test for connectedness using mapsof intervals.(This is where the actual lecture ended.)29Definition 6.9. In a space X with x, y ∈ X, a path from x to y is acontinuous map f : [0, 1] → X with f(0) = x and f(1) = y . A path from xto y is an arc if it is one-to-one.A space is path-connected if any two points can be joined by a path. Aspace is arc-connected if any two distinct points can be joined by an arc.No proof is given for the fol l owing.Proposition 6.10. The image of a path-connected space under a continuousmap is path-connected, and the image of an arc-connected …