This website is a collection of materials I find interesting and useful. Most of them are based on the coursework I did.

Hope you may find some of these entries helpful as well.

# Welcome

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## Hatcher

Topology of Numbers: Essentials and Solutions > Chapter 0 Preview

Pythagorean Triples Pythagorean triple is a triple of positive integers such that We are looking for all possible solutions of equation (1). A primitive Pythagorean triple is the one whose numbers have no common factor, which is equivalent to saying that no two of them have a common factor, or they are pairwise relatively prime. …

View full post## Primitive Roots Modulo the First 1000 Numbers

Integer Number of Primitive Roots Primitive Roots 2 1 1 3 1 2 4 1 3 5 2 2 3 6 1 5 7 2 3 5 9 2 2 5 10 2 3 7 11 4 2 6 7 8 13 4 2 6 7 11 14 2 3 5 17 8 3 5 …

View full post## Primitive Roots Modulo the First 100 Prime Numbers

Prime Number Number of Primitive Roots Primitive Roots 2 1 1 3 1 2 5 2 2 3 7 2 3 5 11 4 2 6 7 8 13 4 2 6 7 11 17 8 3 5 6 7 10 11 12 14 19 6 2 3 10 13 14 15 23 10 5 …

View full post## 2000 Munkres

Topology: Essentials > Chapter 4 Countability and Separation Axioms

Munkres, Section 30 The Countability Axioms First countability axiom: for every point there is a countable basis at . is called first-countable. Continuous functions and converging sequences in first-countable spaces (compare to §21): Converging sequences of points and the closure (The Sequence Lemma): let be a topological space, then in any topological space: if there is …

View full post## 2000 Munkres

Topology: Solutions > Chapter 4 Countability and Separation Axioms

Munkres, Section 30 The Countability Axioms 1 (a) Let be a countable basis at . Then for each there is a neighborhood of such that does not contain . Therefore, there is that does not contain . Thus, . (b) Given any and there must be a neighborhood of that does not contain , therefore, …

View full post## 2000 Munkres

Topology: Essentials > Chapter 3 Connectedness and Compactness

Munkres, Section 23 Connected Spaces A connected space is one that cannot be separated into the union of two disjoint nonempty open sets. Otherwise such a pair of open sets is called a separation of . Connectedness is a topological property: any two homeomorphic topological spaces are either both connected, or both disconnected, and the same …

View full post## 2000 Munkres

Topology: Solutions > Chapter 3 Connectedness and Compactness

Munkres, Section 23 Connected Spaces 1 If is disconnected then there is a separation in the topology which is also separation in . Equivalently, if is connected then so is . 2 If is disconnected then there is a separation of the union and each set of the sequence lies within either or . Suppose …

View full postTopology of Numbers: Essentials and Solutions > Chapter 0 PreviewPrimitive Roots Modulo the First 1000 NumbersPrimitive Roots Modulo the First 100 Prime Numbers2000 Munkres

Topology: Essentials > Chapter 4 Countability and Separation Axioms2000 Munkres

Topology: Solutions > Chapter 4 Countability and Separation Axioms2000 Munkres

Topology: Essentials > Chapter 3 Connectedness and Compactness2000 Munkres

Topology: Solutions > Chapter 3 Connectedness and Compactness

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