Before diving into the solutions, it's essential to understand the textbook itself and why it has remained a popular choice for over half a century. Written by Bert Mendelson, a former Professor of Mathematics at Smith College, the book was originally conceived as a set of lecture notes for a one-semester undergraduate course. Its principal aim is to provide a simple, thorough survey of elementary topics to students whose preparation includes a calculus sequence where some attention has been paid to definitions and proofs of theorems.
: Offers step-by-step explanations for specific sections, particularly for Chapter 1 [6]. Textbook Content Overview
: The book is widely considered a very "gentle" start to the subject, offering better intuition for people coming in without as much knowledge of analysis. One reviewer on Amazon notes that the pedagogy was excellent and the development of topics "made sense" in going from metric spaces (a notion that is generally more intuitive) to abstract topological spaces. Introduction To Topology Mendelson Solutions
Mendelson’s exercises are notoriously "dense." A typical problem might read: "Let X be a topological space. Prove that the closure of a set A equals the intersection of all closed sets containing A." This is a one-line proof in your head, but a beginner might spend 30 minutes formalizing it.
Let $X$ be a compact topological space and let $f: X \to Y$ be a continuous function. Let $U_\alpha$ be an open cover of $f(X)$. Then, $f^-1(U_\alpha)$ is an open cover of $X$. Since $X$ is compact, there exists a finite subcover $f^-1(U_\alpha_i)$. This implies that $U_\alpha_i$ is a finite subcover of $f(X)$, and hence $f(X)$ is compact. Before diving into the solutions, it's essential to
: Certain printings (e.g., Allyn & Bacon) have been noted to include full solutions or substantial hints for the majority of questions.
The distance function $d(x,y)$ and what "closeness" means. Mendelson’s exercises are notoriously "dense
Heavy utilization of the triangle inequality to bound distances and prove limits.