Topological spaces - construction and purpose - Lec 04 - Frederic Schuller

The video begins with an introduction to topological spaces, emphasizing their foundational role in understanding concepts like convergence and continuity. The presenter outlines the historical development leading to the definition of a topological space, which is presented as a set M with a topology defined by a subset of the power set of M. Three conditions are required for a topology: the empty set and the set itself must be included, the intersection of any two subsets in the topology must also be in the topology, and the union of any collection of subsets in the topology must be in the topology.

The presenter discusses the variety of topologies that can be defined on a single set, highlighting that unless the set contains only one element, there are multiple possible topologies. The number of possible topologies for sets with up to seven elements is given, and the concept of the discrete topology and the chaotic topology is introduced. The former includes every possible subset as open, while the latter includes only the empty set and the set itself as open.

Several examples of topologies are given, including the discrete topology on any set, the topology on a three-element set, and the standard topology on real numbers. The presenter explains that the standard topology on real numbers (or R^D) is constructed in three steps, starting with defining open balls for every point in the space and every positive radius. The standard topology is crucial for understanding convergence and continuity in the context of real numbers.

The construction of the standard topology on R^D (real numbers to the Dth power) is explained in detail. The presenter defines open balls using the two-norm and shows how these open balls can be used to determine whether a set is open in the standard topology. The video also illustrates how open sets can be visualized in two dimensions and how this concept extends to higher dimensions.

The process of constructing new topologies from given ones is discussed, with a focus on the induced topology on a subset. The presenter explains that if M is a topological space and N is a subset of M, the induced topology on N can be defined by intersecting every open set in M with N. This construction is shown to satisfy the axioms of a topology, and the presenter emphasizes the importance of this technique in building new topologies from existing ones.

The concepts of open and closed sets within a topological space are explored. The presenter clarifies that a set is considered closed if its complement with respect to the whole set is open. It is noted that in general, a subset of a topological space can be open, closed, both, or neither. The special cases of the empty set and the entire set are highlighted as being both open and closed in any topological space.

The presenter provides a real-life example of using an induced topology, specifically on a circle. Two methods are discussed: defining the circle as a subset of R^2 with the standard topology induced on it, and understanding the circle as a quotient space of the real line with an equivalence relation identifying points that differ by a 2π interval. The video concludes with an assignment for the audience to explore the inherited topology on a quotient space.

The concept of inheriting a topology from two spaces A and B is introduced, focusing on the Cartesian product of these spaces. The product topology is defined and characterized by the condition that a subset U of the Cartesian product A x B is open if and only if for every point in U, there exist open sets in A and B such that their Cartesian product lies entirely within U. The video also touches on the standard topology on R^D and its relation to the product topology.

The presenter remarks on the versatility of topology beyond geometric contexts, highlighting its use in number theory as an example. The video mentions a topological proof by Firstenberg that demonstrates the infinitude of prime numbers using a cleverly chosen topology on the integers. This example showcases the abstract nature of topology and its wide applicability.

The notion of convergence of sequences in topological spaces is defined. A sequence is said to converge to a point if for every open neighborhood of that point, there exists a natural number after which all terms of the sequence fall within that neighborhood. The video provides examples of convergence in the context of the chaotic topology, where every sequence converges to every point, and the discrete topology, where only eventually constant sequences converge.

The presenter discusses convergence in metric spaces, noting that while every topology induced by a metric leads to a general notion of convergence, not every topology is derived from a metric. The example of a sequence in R that converges under the standard topology but not under the discrete topology is given to illustrate this point.

The video takes a half-serious tone to discuss the concept of convergence in the context of scientific theories and research proposals. It cautions against making claims about theories converging to the truth without a well-defined topology on the space of all theories, highlighting the complexity and potential non-transitivity of such claims.

The concept of continuity in topology is defined, stating that a function between two topological spaces is continuous if the pre-image of every open set in the target space is an open set in the domain. The video provides several examples to illustrate this definition, including the case where the domain has the discrete topology, making every function continuous.

The video introduces the concept of a homeomorphism, which is a continuous bijective map with a continuous inverse, implying that the topological spaces are not just set-theoretically isomorphic but also topologically isomorphic. The presenter explains that homeomorphisms are the structure-preserving maps in topology and that two homeomorphic spaces share the same topological structure.

The presenter discusses the challenge of classifying topological spaces, noting that unlike set theory where cardinality provides a complete classification, no such complete set of properties is known for topological spaces. The video emphasizes the importance of recognizing open problems in the field and suggests that the lack of a classification for topological spaces is an open question in the field of topology.