Real analysis Part 1_ Basic Topology

Explore fundamental set concepts in real analysis, including finite and infinite sets, equal and subset relations, singleton and power sets, and the basics of the functions and the relations.

Explore the basic concepts of relations and functions, including the Cartesian product, domain, range, images, and preimages; learn function criteria, well-definedness, and injective and surjective mappings.

Define countable and uncountable sets through mappings with natural numbers; show rationals and integers are countable, while the real numbers are uncountable.

Show that every infinite subset of a countable set is countable, with finite subsets also countable, and illustrate with primes, rationals, and uncountable reals.

Prove that the real numbers in (0,1) are uncountable via Cantor’s diagonal argument, and show the irrational numbers are uncountable, while the union of countable sets is countable.

Demonstrate that the union of countable sets is countable by listing elements in a two-dimensional sequence and establishing a one-to-one correspondence with the natural numbers.

Explore algebraic numbers as real roots of polynomials with integer coefficients, and transcendental numbers like e and pi, showing algebraic numbers are countable while transcendental numbers are uncountable.

Define a metric space as a set X with a distance D that is positive, symmetric, and satisfies the triangle inequality. Illustrate with D(x,y)=|x−y|/(1+|x−y|) and D* = D/(1+D).

Explore constructing multiple metrics on metric spaces, including infinite D star variants, the max metric on R^2, and sup and integral metrics on function spaces.

Explore norms and the metric induced by a norm, including positivity, zero element, symmetry, Minkowski inequality, and the triangle inequality. See function-space examples with the integral of squared differences.

The lecture defines neighborhoods and interior points, shows open intervals, and proves that every real number is an interior point of R while Z, Q, and R\Q have none.

Define open sets and interior points, showing every point has a neighborhood inside the set, with examples like R, R\Z, the empty set, and Z, Q, R\Q.

Study open sets by neighborhoods and interior points, showing unions and finite intersections are open, infinite intersections may fail, and interior is contained in the set, but not vice versa.

Explore De Morgan's law for set operations: prove that the union of a family of sets' complements equals the intersection of their complements, and vice versa.

Define a limit point as a point whose every neighborhood contains a distinct point of E. Explore open intervals, endpoints, and the density of rationals and irrationals, and Z.

A closed set contains all its limit points. The rationals are not closed, while the real numbers, integers, natural numbers, and the empty set are closed.

Study the derived set as the set of limit points and its relation to closure in real numbers. A limit point yields infinitely many points of E in every neighborhood.

Explore the open set and closed set theorem, proving that a finite set has no limit point, and that a set is open iff its complement is closed.

Prove that a set is closed if and only if its complement is open, using neighborhoods and limit points to establish both directions.

The lecture shows that the intersection of any collection of closed sets is closed. Finite unions of closed sets are closed; infinite unions may not be, F_n = [-1+1/n,1-1/n] yields (-1,1).

Demonstrate that the closure Ē is closed by showing any limit point of Ē belongs to Ē, using neighborhoods, limit points, and the derived set E′.

Prove that in a metric space, E equals its closure if and only if E is closed, using limit points and the smallest closed set containing E.

Define bounded above and bounded below, identify upper and lower bounds, and explain least upper bound and greatest lower bound with examples like {0,1,2,3,4,5} and sine x.

Learn the bounded set concept in real analysis, with examples of limit points, least upper bound, and the role of closure and neighborhoods.

Explore the discrete metric space where d(x,y)=1 for distinct points, making every subset open and closed, and note that discrete spaces have no limit points.

Explore isolated points and limit points, distance between sets, and closures to determine when a set is dense in another, with rationals in reals and integers as examples.

Explore open sets relative to a subspace: E equals Y ∩ G for some open G in X, with neighborhoods intersected with Y contained in E.

the lecture defines open covers and finite subcovers and explains compact sets, then uses the rationals in (a, b) to illustrate a closed, bounded set that is not compact.

Show that a subset is compact relative to a subspace if and only if it is compact relative to the ambient space, via open covers and finite subcovers.

The lecture shows compact subsets of matrix space are closed by proving their complement is open, using disjoint neighborhoods and a finite subcover.

Demonstrate that closed subsets of a compact set are compact and that the intersection of a closed subset with a compact set is also compact, via open covers.

Demonstrate that in a metric space, a collection of compact subsets with the finite intersection property has nonempty intersection; proof uses open covers and finite subcovers via de Morgan's law.

presents a corollary: a nested sequence of nonempty compact sets G_n with G_{n+1} ⊆ G_n has nonempty intersection. The proof follows the previous theorem's idea using compactness.

Shows that every infinite subset of a compact set has a limit point in the set by a contradiction using open covers and finite subcovers.

Present the equivalence between closed and bounded sets and compactness, and show that every infinite subset has a limit point, as discussed in the lecture.

Explore the equivalence between compactness and closed, bounded sets by proving that every infinite subset has a limit point in E, using neighborhood and contradiction arguments and linking to Heine-Borel.

The Heine-Borel theorem links closed and bounded sets to compactness. The lecture proves both directions, using limit points and covers, to show compactness implies closedness and boundedness, and vice versa.

Present the Weierstrass theorem: every bounded infinite subset of R^k has a limit point, using open-cover argument; it notes that closed and bounded sets in R^k are compact.

A perfect set is a closed subset where every point is a limit point, with no isolated points. Examples include the real numbers, while rationals are not perfect.

Prove that a nonempty perfect set B is uncountable by showing every nested closed neighborhood intersection is nonempty, using closure, compactness, and a contradiction if B were countable.