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Mathematics · 400 Level
This is the complete lecture note for MTH 405 General Topology, prepared for Mathematics students at 400 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS MTH 405: General Topology Series 1: Foundations of Topological Spaces 1.1 Definition of Topological Spaces 1.1.1 Concept of a set and topology 1.1.2 Examples of topological spaces 1.1.3 Motivation for studying topology 1.2 Open and Closed Sets 1.2.1 Definition of open sets 1.2.2 Definition of closed sets 1.2.3 Relationship between open and closed sets 1.3 Neighbourhoods 1.3.1 Definition of neighbourhood of a point 1.3.2 Neighbourhood systems 1.3.3 Role of neighbourhoods in topology 1.4 Coarser and Finer Topologies 1.4.1 Definition of coarser topology 1.4.2 Definition of finer topology 1.4.3 Examples and comparisons Introduction When studying mathematics, one often begins with familiar structures such as sets and functions. However, as we move into the field of topology, we encounter a new way of thinking about spaces, one that focuses not on distances but on the relationships between points and the openness of sets. This perspective is fundamental to modern analysis and provides the language for many advanced areas of mathematics. By starting with the foundations of topological spaces, you will be equipped to understand the deeper concepts that follow in this course. The aim of this Series is to introduce you to the basic building blocks of topology, including the definition of topological spaces, the role of open and closed sets, neighbourhoods, and the comparison of coarser and finer topologies. This will give
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS you the essential framework needed to progress confidently through the rest of the course. We shall commence this Series by defining what a topological space is and examining simple examples to build intuition. Then, we will study open and closed sets, clarifying how they interact and why they are central to topology. Following that, we will explore neighbourhoods, which provide a local perspective on points within a space. Finally, we will wrap up by comparing coarser and finer topologies, showing how different choices of topology can alter the structure of a space. 1.1 Definition Of Topological Spaces Topology begins with the idea of a topological space, which is a set equipped with a collection of subsets that satisfy certain rules. A topological space is defined as a pair (X,T), where Xis a set and Tis a collection of subsets of Xcalled a topology. The subsets in Tare known as open sets, and they must satisfy three conditions: the empty set and the whole set Xare open, the union of any collection of open sets is open, and the intersection of any finite number of open sets is open. Fill in the blank: A topological space is a pair (X,T), where Tis a collection of __________ sets. 1.1.1 Concept of a set and topology A set is simply a collection of distinct objects, often called elements. A topology on a set is a way of specifying which subsets of the set are considered open. This structure allows us to study continuity, convergence, and connectedness without relying on distances.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS set with a topology defined by open subsets. Fill in the blank: A topology on a set specifies which subsets are considered __________. 1.1.2 Examples of topological spaces Discrete topology: Every subset of Xis open. Trivial topology: Only the empty set and the whole set are open. Standard topology on R: Open intervals form the basis of the topology.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS table showing examples of discrete, trivial, and standard topologies Topology Type Formal Definition / Open Sets Key Characteristic Typical Example Trivial (Indiscrete) \tau = \{\emptyset, X\} The "coarsest" possible topology; only the empty set and the entire space are open. On X = \{1, 2, 3\}, the open sets are just \{\emptyset, \{1, 2, 3\}\}. Discrete \tau = \mathcal{P}(X) (The Power Set) The "finest" possible topology; every subset of X is considered an open set. On X = \{a, b\}$, the open sets are \{\emptyset, \{a\}, \{b\}, \{a, b\}\}$ Standard (Euclidean) \tau = \{U \subseteq \mathbb{R} \mid U \text{ is a union of open intervals } (a, b)\} Based on the concept of distance (metric); open sets are defined by intervals without endpoints. On the real line \mathbb{R}, any interval like (0, 1) or (3, 5) \cup (10, 12) is open. Fill in the blank: In the discrete topology, __________ subset of the set is open. 1.1.3 Motivation for studying topology Topology provides a framework for generalising concepts such as continuity and convergence beyond metric spaces. It allows mathematicians to study spaces in terms of their structure rather than their geometry. This abstraction is essential in analysis, algebra, and geometry. Fill in the blank: Topology studies spaces in terms of their __________ rather than their geometry.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Pilot questions 1.1 A topological space is defined as: A. A set with a collection of open sets satisfying certain rules B. A set with only closed sets C. A set with random subsets D. A set with a metric only In the discrete topology: A. Every subset is open B. Only the empty set is open C. Only the whole set is open D. No set is open The trivial topology consists of: A. The empty set and the whole set B. All subsets of the set C. Only singletons D. Open intervals The standard topology on Ris generated by: A. Open intervals B. Closed intervals C. Singletons D. Random subsets Topology studies spaces in terms of: A. Structure B. Geometry only C. Randomness D. Distance only 1.2 Open And Closed Sets In topology, the concepts of open sets and closed sets are fundamental. An open set is a subset of a topological space that belongs to the topology, meaning it satisfies the rules defined for openness. A closed set is the complement of an open set within the space. These two notions are not opposites in the everyday sense, but rather complementary: a set can be both open and closed, or neither, depending on the topology. Fill in the blank: A closed set in a topological space is defined as the __________ of an open set. 1.2.1 Definition of open sets An open set is any subset of a topological space that is included in the topology. For example, in the standard topology on R, open intervals such as (a,b)are open sets. Open sets are used to define continuity and convergence in topology.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Open intervals as open sets in the standard topology on R. Fill in the blank: In the standard topology on R, sets of the form (a,b)are __________ sets. 1.2.2 Definition of closed sets A closed set is the complement of an open set. For example, in the standard topology on R, closed intervals such as [a,b]are closed sets. Closed sets are important in defining concepts such as compactness and closure.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Closed intervals as closed sets in the standard topology on R. Fill in the blank: In the standard topology on R, sets of the form [a,b]are __________ sets. 1.2.3 Relationship between open and closed sets Open and closed sets are related through complementation. If a set is open, its complement is closed, and vice versa. Some sets can be both open and closed,
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