LECTURE NOTES FOR TOPOLOGY I

General topology

The category of topological spaces

Topological spaces
First examples
Continuous maps and homeomorphisms

1.2) The closureA is the smallest closed subset containing A. 1.3) The closureA is the intersection of all closed subsets containing A:. Define a topology in the following way: let B be the set of all finite intersections of elements of S (including the empty intersection, which consists entirely of X). Then you see that each card p:X //Sδ is face up; However, for pto to be continuous, for anys∈Sthefiberovers - that is, the inverse image f−1{s} - would have to be an open set.

This suggests something that may seem surprising at first: topology does not "see" the difference between an infinitely long line and an open interval of finite length.

Constructing topologies and continuous maps

Quotients
Coverings and locality

Since Bx is an open system of neighborhoods of x∈X, there is an elementV ∈ Bx such thatV ⊂U. Then (with the axiom of choice) one can choose for each pointx∈U one elementVx∈ Bx, such that Vx⊂U. The previous exercise can be used to show that the Sorgenfrey line is finer than the standard (=order) topology onR.

The Euclidean spaces are not compact; the open spheresBn(x,"") do indeed form an open cover, but there is no finite partial cover.

Compactness I

Compact spaces
The Tychonoff Product Theorem
The compact-open topology

Hint: if U is an open set of X not contained in M, show that there is a finite collection {Vj}mj=1 of elements of M such that {U} ∪ {Vj}mj=1 is a coverage of X.). Okay, now it's time to consider our subbases; again, we assume that every civil subset of S is irrational. Show that X is compact (with the order topology) if and only if it is order complete.

Prove that a sequence fn of continuous functions X //Y converges uniformly if and only if it converges to C(X,Y) with a compact open topology. We have already seen that for subspaces of Euclidean space, compactness is equivalent to the Heine–Borel property (closed and bounded) and the Bolzano–Weierstraß property (the property that every sequence has a convergent subsequence). In this section, we will discuss analogies and generalizations of these concepts, and we will also talk about the concept of anetorMoore–Smith sequence.

4.1.1.1) X is said to be limit point compact if and only if every infinite subsetAafX has a limit point (ie a point x∈X contained in A− {x}). X is said to be countably compact if and only if every countable open cover has a finite subcover. These three concepts seem remarkably similar, and in Rn (and more generally, in any metrizable space) they are fuzzy and, moreover, correspond to compactness.

Show that a space X is countably compact if and only if every nested sequence X ⊃V1⊃V2 ⊃ V3⊃ · · · of closed, nonempty subspaces of X has a nonempty intersection. If this choice is impossible at any (finite) stage, then we have our required finite subsheet;.

Compactness II

Forms of compactness
Locally compact Hausdorff spaces

Show that for any ordinalα, the order topology and the discrete topology coincide if and only ifα contains no limit ordinal other than 0. Therefore, the spaceω1 (when equipped with the order topology) is sequentially compact, but it cannot be compact since it contains no maximal element. A space X is said to be Hausdorff if, for two distinct points x,y∈X, there are neighborsU afxogV of such thatU∩V.

A space Every subspace of a Hausdorff space is Hausdorff, and every product of a set of Hausdorff spaces is Hausdorff. The set{Wy |y∈K} is an open covering of K, so there is a finite set of pointsy1,y2,.

Show that for every point x ∈X −K there exists a neighborhoodU of x and an open set V containing K such that U∩V=∅, and the closureV is compact. If that is not the case, then problem 32 gives for each pointx∈X−U an open neighborhoodVxofxand an open setWxwhose closure is so compact that. I can apply the previous lemma to get an open neighborhoodW ofx, such that W is compact and is inV.

We now turn to the Alexandroff compactification at a point of a spaceX. Then define X? to be setXt {∞}, with the following topology: a set U ⊂X? will be said to be open if and only if U is an open subset of X, or elseU contains∞, and X−U is closed and compact. Then the neighborhood of a subset A⊂X is a set whose interior contains A. Let X be a space and let A,B⊂X be two subspaces of X. 5.1.3.1) The setAandBare is said to be disjoint if both A∩BandA∩Bare are empty.

Countability and separation

Taxonomy of separation
Normality, Urysohn’s lemma, and the Tietze extension theorem

A space X is said to be irreducible (or hyperconnected) if no two non-empty open subsets of X are disjoint. Suppose D⊂R>0 is a dense subset, and suppose that for each element t∈D there is a subset Ft of a set X such that. It is sufficient to check whether the inverse image of the sets(−∞,s)and(−∞,s]are open and closed (respectively) for each s∈R.

The previous lemma shows that the inverse images of sets of the first kind, (−∞,s), are the union of open sets, hence open. It now follows from the previous lemma that the inverse images of sets of the second kind, (−∞,s], are closed (and so we are done) if the following equality is obtained: It is clear to see that the conditions of the previous lemma applies, and this constructs our continuous function f.

X is said to satisfy the first axiom of countability or to be first-countable if for every point x∈X there is a countable fundamental system of neighborhoods ofx. The space X is said to satisfy this second axiom of countability or to second-countable if the topology onX has a countable basis. Any subspace of a first-countable (respectively, second-countable) space is first-countable (resp., second-countable), as is any countable product of first-countable (resp., second-countable) spaces.

Give an example showing that an uncountable product of second-countable spaces does not have to be first-countable. Show that for every subsetA⊂X there is a pointx∈X in the closureAif, and only if there exists a set of points ofAconverging tox.

Metrization theorems

First and second countability
Urysohn’s metrization theorem
The Bing–Nagata–Smirnov metrization theorem

It is not difficult to see that the Hilbert cube is separable and metric; so also every subspace. Therefore, we still need to show that the space aT3, which satisfies the second axiom of countability, is homeomorphic to a subspace of the Hilbert cube. Using the above embedding lemma, it suffices to construct a countable family F of continuous functions X //[0, 1] such that for every closed subset A⊂X and any point x∈X−A there exists an elementf ∈ F such that f(x) ∈/ f(A).

This is a countable set of functions, and it remains to be shown that it has the desired properties. Fill in the details I left out in the proof of Urysohn's metrization theorem, and write a complete proof. A family V of subspaces of a space .

I can show that if a space satisfies some abstract conditions, then it can have a metric that gives rise to its topology. Let us list some of the purely topological consequences of the metrizability of a spaceX. 6.3.6.2) Dual, for any subspace A⊂ X, a point x ∈X is a member of the closure Aif if and only if there is a sequence of points iA that converges to A. 6.3.6.4) A subspace A⊂X is compact if and only if it is countably compact if and only if it is sequentially compact.

It is not difficult to see that if I have a continuous functionf onU, then I can restrict it to a continuous function f|V on any open subsetV ⊂U. I could try the same trick with any "target" topological spaceY: for any open setU ∈ Op(X), write

Sheaf theory

Presheaves
Stalks
Sheaves
The espace étalé and sheafification
Direct and inverse images of sheaves

Amorfismor simple map of presheavesφ:F //G is the following data: for any open set U ∈ Op(X), a map (called the component ofφ). Given morphisms of presheavesφ:F //G andψ:G //H, we can form the compositeψ◦φ:F //H in the following way: for any open setU ∈ Op(X), set. 49.3) Finally, show that a morphism of presheavesφ:F //G is an isomorphism if and only if, for any open set U∈ Op(X), the componentsφU are all bijective.

Show that for any space X and any pair of open sets U,V ∈ Op(X) with V ⊂U there exists a unique presheaf morphism hV //hU. Show that for any space X, any open set U∈ Op(X) and any prebundle F on X there exists a natural bijection. Show that for any two spaces X and Y, the presheaf OXY of functions with local Y values is isomorphic to the presheaf Γ(Y×X/X) of the local sections of the projection map Y×X //X.

For any pointx∈X, we can easily compute the stemΓ(Y/X)x: there exists an open neighborhood Uofx whose inverse image is homeomorphic (via p) to a disjoint union of copies of U. In other words, an equalizer is if and only if the following condition is satisfied: for each α∈Λ there is a section α∈ F(Uα) such that the restrictions for any pairβ,γ∈Λ. A presheafF on a spaceX is said to be asheaf if the graph of any open setU ∈ Op(X) and any open cover{Uα}α∈ΛafU.

Show that the natural morphism pF for any spaceX and any preheafF onX: ´Et(F) //X is a local homeomorphism. In other words, for any morphism F //G there exists a unique morphism aF //G such that the diagram.

An introduction to homotopy theory

Covering space theory

Connectedness and π 0
Covering spaces and locally constant sheaves
The étale fundamental group
Homotopy classes of loops and lifts
Universal covering spaces

Computing the étale fundamental group

The étale fundamental group of S 1
Seifert-van Kampen Theorem
Simple connectedness

Homotopy theory

Paths and π 0
Homotopies of maps
The homotopy-theoretic fundamental group
Homotopy invariance of the étale fundamental group

The comparison theorem

The two fundamental groups

Note thatπét0X is a set with a continuous mapX //(πét0X)δ such that for any setSand any continuous mapX //Sδ there exists a unique mapπét0X //S such that the diagram. In particular, the definition is non-constructive in the sense that it is not clear from the definition alone that πét0X exists for a given spaceX. Show that if X is a space such that πét0X exists, then the continuous map X //πét0X is surjective.

Conclude that πét0X exists if and only if there exists a maximal undiscovered decomposition of X, that is, a decomposition of X such that every sum is connected. 63.3) the setGLnRofn×invertible matrices with real entries, with the subspace topology of the inclusion GLnR⊂Rn2, forn>0,. The constructionX //πét0X is functorial in the sense that for any continuous map f :X //Y, there is a corresponding mapπét0 f :πét0X //πét0Y if both πét0X andπét0Y exist .

In fact, the definition implies that there exists a unique mapπét0f :πét0X //πét0Y such that the diagram. If πét0X exists, then we say it is of finite type, ifπét0E exists and the fibers of πét0p:πét0E //πét0X are finite. Then a universal locally constant envelope of (X,x) is a locally constant envelopeEonX together with a kernel of a sectionυ∈ Fx such that for every locally constant envelopeF and every kernelσ∈ Fx there is a unique morphismφ:E //F of slices such as. thatφx(υ) =σ.

Show that the universal locally constant sheaf (E,υ)on (X,x), if it exists, has the following property: for any locally constant sheafF onX, the map MorX(E,F) //Fx, φ // φx (υ), is a bijection. Deduce that if there exists a universal locally constant cut on(X,x), then it is unique up to a unique isomorphism.