Summary of "Мищенко А. С. - Введение в топологию - 1. Основные определения теории множеств"
Summary of the Lecture: “Мищенко А. С. - Введение в топологию - 1. Основные определения теории множеств”
This lecture provides an introduction to set theory concepts foundational for topology, outlining key definitions, set operations, mappings, cardinalities, and related theorems. It serves as a preparatory overview for further study in topology and related mathematical fields.
Main Ideas and Concepts
1. Course Overview
- The course lasts six months and summarizes topology-related content from mathematical analysis, algebra, and analytic geometry.
- It prepares students for advanced topics like functions of a complex variable.
- Course topics include:
- Language of set theory
- Metric and topological spaces
- Continuous mappings
- Logical spaces and axioms
- Compact spaces
- Gamma theory, currents, and homology theory
- Recommended textbooks:
- Avital Vitalich Fedorchuk’s Introduction to Topology
- Elementary Topology by a team from St. Petersburg University
- Classic book by Pavel Sergeyevich Aleksandrov on set theory and general topology
2. Language of Set Theory
- Sets are considered as undefined primitive concepts.
- Elements of sets are denoted by lowercase letters (e.g., ( x \in X )), where ( X ) is a set.
- Sets can be defined by properties of their elements, but care must be taken to avoid contradictions.
- Examples of sets: natural numbers, integers, rational numbers, real numbers, functions on intervals.
- Subsets are always relative to a larger set; there is no absolute subset.
3. Basic Set Operations
- Intersection (( A \cap B )): elements common to both sets.
- Union (( A \cup B )): elements in either set.
- Set difference (( B \setminus A )): elements in ( B ) but not in ( A ).
- Complement of a set ( A ) in ( X ) is ( X \setminus A ).
- De Morgan’s laws apply to complements, unions, and intersections.
- These operations extend to indexed families of sets ( {A_\alpha}_{\alpha \in I} ) with union and intersection over indices.
4. Mappings (Functions)
- A mapping ( f: A \to B ) assigns each element of ( A ) to exactly one element of ( B ).
- Mappings can be defined by formulas or logical conditions.
- Composition of mappings ( g \circ f ) is defined when the codomain of ( f ) matches the domain of ( g ).
- Identity mapping ( \mathrm{id}_A: A \to A ) maps each element to itself.
- Important types of mappings:
- Injection: different elements of ( A ) map to different elements in ( B ).
- Surjection: every element of ( B ) is an image of some element of ( A ).
- Bijective (one-to-one correspondence): both injective and surjective.
- Inverse mapping ( f^{-1} ) exists only for bijections.
5. Equivalence of Sets by Cardinality (Power)
- Sets ( A ) and ( B ) have the same cardinality if there exists a bijection between them.
- The cardinality of a set ( A ) is denoted ( |A| ) or ( \mathrm{card}(A) ).
- Finite sets have cardinality equal to the number of elements.
- Infinite sets exist, e.g., the set of natural numbers ( \mathbb{N} ).
- Countable sets: finite or bijective with ( \mathbb{N} ).
- Uncountable sets exist; Cantor’s theorem demonstrates this.
6. Cartesian Products and Graphs of Mappings
- Cartesian product ( A \times B ) is the set of ordered pairs ( (a,b) ) with ( a \in A ), ( b \in B ).
- The graph ( \Gamma_f \subset A \times B ) of a mapping ( f ) consists of pairs ( (a, f(a)) ).
- Cartesian products can be generalized to indexed families ( \prod_{\alpha} A_\alpha ).
- Existence of elements in infinite products relates to the Axiom of Choice.
7. Disjoint (Unrelated) Unions
- Unrelated union ( A \sqcup B ) assumes ( A \cap B = \emptyset ).
- This concept generalizes to families of sets.
8. Images and Preimages of Sets under Mappings
- Image of a subset ( C \subset A ) under ( f ) is [ f(C) = { f(a) \mid a \in C } \subset B. ]
- Preimage of a subset ( D \subset B ) under ( f ) is [ f^{-1}(D) = { a \in A \mid f(a) \in D }. ]
- For surjective mappings, preimages have useful properties.
9. Factor Sets (Quotients)
- Given ( B ) and a subset ( A \subset B ), the factor set ( B / A ) identifies all elements of ( A ) as a single point.
- This is a basic construction before adding additional structure (e.g., topology).
10. Cardinality and Cantor’s Theorem
- The power set ( 2^A ) is the set of all subsets of ( A ).
- Cardinality of ( 2^A ) is strictly greater than that of ( A ) (Cantor’s theorem).
- This shows the existence of uncountable sets.
- Bernstein’s theorem: If there are injections ( A \to B ) and ( B \to A ), then ( |A| = |B| ).
- The lecture promises to prove these theorems in the next session.
Methodology / Key Definitions and Instructions
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Defining sets by properties: [ X = { a \in Y \mid P(a) }, ] where ( Y ) is a known set and ( P ) a property.
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Set operations:
- Intersection, union, difference, complement.
- Use indexed families for unions and intersections.
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Mappings:
- Define ( f: A \to B ) by formulas or logical conditions.
- Composition: ( g \circ f: A \to C ).
- Identity mapping: ( \mathrm{id}_A(a) = a ).
- Injection, surjection, bijection.
- Inverse mapping ( f^{-1} ) for bijections.
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Cartesian product: [ A \times B = { (a,b) \mid a \in A, b \in B }. ]
- Graph of ( f ): [ \Gamma_f = { (a, f(a)) }. ]
- Generalized product over indexed families.
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Disjoint unions: [ A \sqcup B \quad \text{when} \quad A \cap B = \emptyset. ]
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Image and preimage:
- Image: ( f(C) ).
- Preimage: ( f^{-1}(D) ).
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Factor set: [ B / A ] identifies all elements of ( A ) as one point.
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Cardinality:
- Define cardinality via bijections.
- Use injections to compare cardinalities.
- Cantor’s theorem: [ |A| < |2^A|. ]
- Bernstein’s theorem: mutual injections imply equal cardinality.
Speakers / Sources
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Speaker: Мищенко А. С. (A. S. Mishchenko) Lecturer presenting the introduction to topology and set theory concepts.
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Referenced authors/textbooks:
- Avital Vitalich Fedorchuk (Introduction to Topology)
- Team from St. Petersburg University (Elementary Topology)
- Pavel Sergeyevich Aleksandrov (Introduction to Set Theory and General Topology)
This summary captures the foundational concepts and prepares the student for further study in topology by grounding them in set theory, mappings, and cardinality.
Category
Educational
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