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MTH251: Set Topology (Spring 25)

anonymous@undisclosed.example.com (Anonymous) — Tue, 29 Apr 2025 10:10:47 +0000

MTH251: Set Topology (Spring 25) [MTH251 Set Topology] Set topology is a branch of mathematics that studies the properties of shapes and spaces that remain unchanged even if they are stretched, twisted, or deformed (without tearing or gluing). It helps us understand concepts like continuity, connectedness, and boundaries.$\mathbb{R}$$T_1$$\mathbb{Z}$$A=\{1,2,3,...,20\}$$\mathbb{R}$$\mathbb{Q}$$\mathbb{R}$$A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$$A$$\mathbb{R}$$A=\mathbb{N}$$B=\{1,2,3,.…

Metric Spaces (Notes)

anonymous@undisclosed.example.com (Anonymous) — Fri, 18 Aug 2023 18:05:58 +0000

Metric Spaces (Notes) [Metric Spaces (Notes)] These are updated version of previous notes. Many mistakes and errors have been removed. These notes are collected, composed and corrected by Atiq ur Rehman, PhD. These are actually based on the lectures delivered by Prof. Muhammad Ashfaq (Ex HoD, Department of Mathematics, Government College Sargodha). $(X,d)$$x,y\in X$$$\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,\,d(x,\,y).$$$A^c$$A\subset X$$x \in X$$B(x;r)$$A \subset X$$f:(X,d)\to (Y…

MTH251: Set Topology (Spring 18)

anonymous@undisclosed.example.com (Anonymous) — Fri, 07 Feb 2025 11:25:49 +0000

MTH251: Set Topology (Spring 18) [Set Topology] Topology is an important branch of mathematics that studies all the “qualitative” or “discrete” properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (infinitely extreme) ones.$\mathbb{R}$$T_1$$\mathbb{Z}$$A=\{1,2,3,...,20\}$$\mathbb{R}$$\mathbb{Q}$$\mathbb{R}$$A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$$A$$\mathbb{R}$$A=\mathbb{N}$$B=\{1,2,3,.…

Khuram Ali Khan

anonymous@undisclosed.example.com (Anonymous) — Fri, 13 Jun 2025 12:03:59 +0000

Khuram Ali Khan Khuram Ali Khan, PhD Associate Professor Department of Mathematics University of Sargodha Sargodha - PAKISTAN. Email: Field of Research: Difference and functional equations, Real functions, Mathematical inequalities involving convex functions, Time Scales Calculus, Soft Sets

Topology: Handwritten Notes

anonymous@undisclosed.example.com (Anonymous) — Sat, 01 Mar 2025 09:43:45 +0000

Topology: Handwritten Notes [House of Tau] A topological space is a collection of points with a topology-a structure that describes how close two points are to one another. It is a generalisation of Euclidean spaces that makes it possible to investigate boundaries, continuity, and connectivity. A topology is a group of open sets, or subsets, that adhere to certain principles.$T_0$$T_1$$T_2$$\varepsilon-$

Chapter 03 - Limits and Continuity

anonymous@undisclosed.example.com (Anonymous) — Sun, 07 Feb 2021 16:49:57 +0000

Chapter 03 - Limits and Continuity * Limit of the function, examples and definition * Theorem: Suppose (i) $(X,{d_x})$ and $(Y,{d_y})$ be two metric spaces (ii) $E\subset X$ (iii) $f:E\to Y$ i.e. f maps E into X (iv) p is the limit point of E. Then $\lim_{x\to p} f(x)=q$ iff $\lim_{n\to\infty}f(p_n)=q$ for every sequence {$p_n$} in E such that ${p_n}\ne p$$\lim_{n\to\infty}{p_n}=p$$\lim_{x\to c}f(x)$$c\in G$$\lim_{x\to c}f(x)=l$$\varepsilon$$\delta>0$$|f(t)-f(s)|<\varepsilon$$\left\{x:|x-c|…

Functional Analysis by Prof Mumtaz Ahmad

anonymous@undisclosed.example.com (Anonymous) — Wed, 18 Sep 2024 18:11:56 +0000

Functional Analysis by Prof Mumtaz Ahmad [Functional Analysis by Prof Mumtaz Ahmad] Functional analysis is a subfield of mathematics that deals with vector space theory and linear algebra. It entails researching the connections between roles, things, incidents, actions, and outcomes. The word $l^\infty$$l^\infty$

Measure Theory by M Usman Hamid & Saima Akram

anonymous@undisclosed.example.com (Anonymous) — Wed, 26 Jun 2024 17:55:59 +0000

Measure Theory by M Usman Hamid & Saima Akram [Measure Theory by M Usman Hamid & Saima Akram] The study of measures on sets is the focus of the mathematical field known as measure theory. A measure is a function that gives specific subsets of a given set a non-negative real integer, indicating their size inferentially. The concept of measure is a formalisation and generalisation of common concepts like mass and event probability as well as geometrical measurements (length, area, and volume).

Multiple Choice Questions (BSc/BS/PPSC) by Akhtar Abbas

anonymous@undisclosed.example.com (Anonymous) — Sun, 24 May 2026 17:45:10 +0000

Multiple Choice Questions (BSc/BS/PPSC) by Akhtar Abbas [Multiple Choice Questions (BSc/BS/PPSC)] These notes are made and shared by Mr. Akhtar Abbas. We are really very thankful to him for providing these notes and appreciates his efforts to publish these notes on MathCity.org. Multiple Choice Questions (MCQs) are given in these notes, which might be helpful in BSc, BS or Punjab Public Service Commission (PPSC) exams.$a$$b$$n$$na > b$$(p − 1)! \equiv −1(mod p)$$p$$p$$p$

General Topology by Azhar Hussain

anonymous@undisclosed.example.com (Anonymous) — Sun, 06 Aug 2023 19:18:17 +0000

General Topology by Azhar Hussain [Topology Notes by Azhar Hussain] The area of topology known as general topology (also known as point set topology) is concerned with the fundamental concepts and constructs of set theory utilised in topology. Most other fields of topology, such as differential topology, geometric topology, and algebraic topology, are built upon it. The three key ideas of point-set topology are connectedness, compactness, and continuity. Continuous functions move points from on…