Lecture 1 (27/jan) - topics Introduction to the course. Definition of topological space; definition of open and closed subsets. The partial order relation of finer/coarser topologies. Examples: the discriete and indiscrete topologies on a set; the topologies on a two-point set; the cofinite topology. Examples of topologies on the real line: the Euclidean topology, the upper semi continuity topology, the lower limit topology. Notion of basis of a topology. A collection of subsets of a set satisfying: 1) it covers the whole space, 2) the intersection of two of them is union of elements of the collection, generates a topology for which it is a basis. Notion of subbasis. Example: the open rectangles and the open balls in R^2 are basis that generate the same topology, called Euclidean topology. The product topology on the Cartesian product of two topological spaces; basis and subbasis for the product topology. References: [Munkres, §12, §13, §15]. --- Lecture 2 (28/jan) - topics The subspace topology. Correspondence between the product of subspaces topologies and the subspace topology of the product. Example: the topology induced by Euclidean R on [0,1]. Definition of interior and closure of subsets. Definition of (open) neighbourhood. Characterization of the points in the closure. Definition of limit point. The closure corresponds to the union of the subset itself with its limit points. Definition of Hausdorff and T1 properties. If a topological space is Hausdorff, then it satisfies the T1 property. Definition of convergence for sequences in a topological space. In a Hausdorff space, a sequence converges to at most one point. Counterexample: a topology on a three-point set such that the constant sequence converges to every point. Definition of continuous function between topological space. Definitions of open map and closed map. For real functions of real variable, the continuity with respect to the Euclidean topology corresponds to the well-known notion of continuity in the analytic sense. Examples of functions which are continuous but not open, or open but not continuous. The identity on a set with different topologiess is continous if and only if the topology on the domain is finer than the topology on the cofomain. Characterization of continuity in terms of basis, closed sets, closure, and the notion of continuity at one point. References: [Munkres, §16, §17, §18] --- Lecture 3 (29/jan) - topics The order topology. Examples: on R with the standard simple order relation, it gives the Euclidean topology; on subsets of R, it gives the induced subspace topology; on N, it gives the discrete topology; R^2 with the lexicographic order; for two copies of N, namely {1,2} x N, it is not the discrete topology because the point 2 x 0 is not open. Exercises. The open balls in R^2 satisfy the axioms to be a basis. When we have two basis on the same set, then an includion between the basis elements will assure a comparison between the induced topologies. In particular, the open balls in R^2 induces the same topology induced by the open rectangles (namely the product topology), which is the Euclidean topology. On R with Euclidean topology, we also have a countable basis, by considering open intervals with rational endpoints. Alexandroff one-point compactification: on R with an \infty point added, we can defined a topology, where R is a subspace. --- Lecture 4 (30/jan) - topics The Golomb topology on Z, to prove that prime numbers are infinitely many. The closure of union of sets coincide with the union of closures; the closure of intersection of sets is only contained in the intersection of closures. Constant functions are continuous; the inclusion of a subspace in a topological space is continuous (in fact, the subspace topology is the coarsest one making the inclusion continous); the composition of two continuous functions is continuous; the restriction of a continuous function to a subspace is continuous; changing the codomain of a continous function to a smaller subspace containing the image remains continuous. Pasting lemma: if we have continous functions defined on open subsets (respectively, on finitely many closed subsets) which merge together appropriately, then the pasting is still continuous on the union. Example: the notions of arc and joint of arcs. Definition and examples of homeomorphism. Definition of embedding. References: [Munkres, §18] --- Lecture 5 (31/jan) - topics In the Euclidean real line, every bounded open interval (a,b) is homeomorphic to (0,1), via a linear function, and also to the whole line R, by using exp and tan. The stereographic projection yields an homeomorphism between the sphere S^n without the north pole and the space R^n, and it extends to a homeomorphism between S^1 and the compactification of R. The set of points where two continuous functions with values in a Hausdorff spaces coincide is closed; in particular, two continuous functions with values in a Hausdorff space that coincide on a dense subset, coincide everywhere. Applications: the only continous functions from R to R that are additive are the linear functions; the cardinality of the space of continuous functions from R to R is equal to the cardinality of R. Any factor X of a product X x Y can be embedded in the product space, by fixing y_0 \in Y and taking the map i_{y_0} : X \to X x Y defined by i_{y_0}(x) = (x, y_0). Any continous function f : X x Y --> Z induces a function f \circ i_{y_0} : X --> Z. The converse does not hold true: there is a function f : R^2 --> R that is not continuous, but when restricted to the x-axis or y-axis is continuous. --- Lecture 6 (03/feb) - topics The box topology and the product topology on the Cartesian product of a (possibly infinite) family of topological spaces. When the product is finite, the box topologogy and the product topology are the same; in general, the product topology is coarser than the box topology. The Cartesian product of topological spaces X_alpha, endowed with either the product or the box topology, is Hausdorff if and only if each factor is Hausdorff (note that the Hausdorff property passes to subspaces and that any factor is embedded in the product topology). We can also prove that the closure of product is the product of closures, with respect to both the product and box topology. The following universal property characterizes the product topology, but does not hol true for the box topology: given a function to a Cartesian product with the product topology, it is continuous if and only if every its components are continuous. Counterexample: if we consider the map f from R to R^N (space of real sequences) endowed with the box topology, sending a real number t to the constant sequence (t,t,....), it is not continuous, even if any component is the identity, so continuous. Definition of metric space. We associate a topology to any metric space (X,d), called the topology induced by d, by taking the collections of open balls as a basis. Example: the discrete metric induces the discrete topology. Any such induced topology is always Hausdorff. In particular, the indiscrete topology on a set with at least two points can not be induced by any metric. We call metrizable a topology which is induced by a metric. Two metrics are called topologically equivalent if they induce the same topology. For example, on R^2, there are three metrics (d_1 given by the sum of the distances of the components, d_2 the usual Eucliden metric, d_\infty given by the max of the distances of the components) that induce the same topology, which corresponds to the product topology on R x R. Fo any metric d, we can define a standard bounded metric by taking \max\{d, 1\}: it is a bound metric, topologically equivalent to d. In particular, boundedness is not a topological property. Reference: [Munkres, §19, §20]. --- Lecture 7 (04/feb, morning) - topics Metrizability of products. R^n is metrizabile (with the product topology). The space R^N of real sequences with the product topology is metrizabile; with the box topology, it is not metrizabile. Sequence lemma: on a metrizable space, we can characterize the closure by sequences; therefore we can also characterize the continuity of a function by sequential continuity. The space R^R of real functions is not metrizable. Reference: [Munkres, §20] --- Lecture 8 (04/feb, afternoon) - topics The uniform metric on the space R^R of real functions. The induced uniform topology is coarser than the box topology and finer than the product topology. Convergence in R^R with the product topology corresponds to pointwise convergence of functions; convergence in R^R with the uniform topology corresponds to uniform convergence of functions. (Recall that the uniform limit of continuous functions is still continuous.) Notion of quotient map. On the quotient of a topological space by an equivalence relation, we define the quotient topology as the unique topology that makes the natural projection a quotient map. Its open sets are characterized as the image of saturated open sets. Examples: the quotient by the sign function; the contraction of the boundary S^1 of a closed disc D^2 to a point gives the sphere S^2; the identification of the parallele edges of a square gives the torus. Reference: [Munkres, §20, §21] --- Lecture 9 (05/feb) - topics Recap on quotient topology. The universal property and a corollary to identify the quotient toplogy. The quotient topology on the quotient of R by the action of Z by translation gives S^1. Note that the Hausdorff and T1 properties do not pass to quotients (counterexample: the quotient of R by the action of Q by translations). Statement: there are conditions assuring the quotient of a Hausdorff space is still Hausdorff (namely, p open map and the relation set R is closed). The quotient of R^n by the action of general linear group of invertible linear maps is a two-point set with one point open, one point closed. The quotient of R^n by the actoin of the orthogonal maps is homeomorphic to [0,+\finty). Reference: [Munkres, §22] --- Lecture 10 (07/feb, morning) - topics Connected spaces. A subset of a topological space is called connected if it is connected with the subspace topology. The continuous image of a connected set is still connected. In particular, being connected is a topological property (for instance, [0,1] can not be homeomorphic to (0,1)). Moreover, if a product of two topological spaces is connected, then each factor is connected, and connectedness is preserved also by taking quotients. The union of a family of connected spaces that intersect pairwise is connected; also the union of a family of connected spaces, each intersecting a fixed one, is connected. Example: S^n is connected. In particular, we define the connected component of a point as the largest connected subset containing that point. A space is connected if and only the connected component of each point is the whole space; in Q, with the induced topology as a subspace of Euclidean R, the connected component of each point q is {q} (in particular, connected components need not to be open). If a subset Y of a topological space is connected, then its closure (and also any subset between it and its closure) is connected too. A topological space is the disjoint union of its connected components; each connected component is closed (but not necessarily open: they are open in case the number of connected components is finite); the relation x~y if and only if the connected component of x and y coincide is an equivalence relation, whose quotient if T1 and totally disconnected. A subset of the real line is connected if and only if it is an interval. In particular, the connected subset of R, up to homeomorphism, are R, [0,+\infty), [0,1]. The intermediate value theorem. Reference: [Munkres, §23-§24]. --- Lecture 11 (07/feb, afternoon) - topics A product space, with the product topology, is connected if and only if each factor is connected (one first proves the case of finite produce by using that the projection map is continuous open and surjective, then one extends to the infinite case by considering a dense subset). The result is no longer true for the box topology (for instance, the space of real sequences with the box topology is disconnected by considering the subsets of bounded and unbounded sequences). A space X is connected if and only if H(X;Z_2), the group of continuous functions with value in Z_2, is isomorphic to Z_2 (constant functions). The definition of the real projective space; it has an open covering with subsets homeomorphic to R^n (in particular, it is a topological manifold: a topological space, Hausdorff and second-countable, such that every point has an open neighbourhood homeomorphic to R^n). There is no continuous function R \to R, sending rational numbers to irrational numbers, and irrational numbers to rational numbers. For any continuos function f : S^n \to R (n\geq 2), there is a point x \in S^n such that f takes the same value on its antipodal, namely f(x) = f(-x); in particular, there is no open subset in R^n homeomorphic to an open subset in R. Reference: [Munkres, §23-§24]. ---