Proving the Completeness of a Metric Space

The complete metric spaces Wikipedia page provides examples of both complete and incomplete metric spaces. These examples have been separated into a new question regarding separable metric spaces, called “Spaces of separable metric spaces II: pointed spaces”. The solution to this is deemed correct. Question: The page on complete metric spaces available on Wikipedia provides […]

Homeomorphism examples

A homotopy between closed curves $gamma_1$ and $gamma_2$ is defined as a continuous map $H:[0,1]times[0,1] to X$ that satisfies $H(0,t) = gamma_1(t)$ and $H(1,t) = gamma_2(t)$ for all $tin [0,1]$. Since homotopies involve continuous deformations, the closed curves $fcircgamma_1$ and $fcircgamma_2$ in $Y$ are homotopic if $f: X to Y$ is a homeomorphism and $gamma_1$, […]

Discrete Topology: An Explanation

When considering a term to distinguish power set topologies from other topologies, we may use the term ‘discrete’ because power sets consist of discrete objects. Therefore, it seems logical to refer to all topologies on sets as ‘discrete’. I was attempting to comprehend why the discrete topology is the most refined/robust form of topology and […]

Generalized Bolzano-Weierstrass Theorem

Given a basis ${v_1,ldots,v_n}$ for $V$, we can consider an isomorphism $T:mathbb{R}^nrightarrow V$ such that $T(e_i)=v_i$. A new norm can be defined on $mathbb{R}^n$ by setting $|x|_{mathbb{R}^n}=|T(x)|_V$. This results in $T$ being a homeomorphism, which maps open balls onto open balls of the same radius, according to the norm on $mathbb{R}^n$. It is important to […]

Exploring the Coexistence of Dense Q in R and a Family of Open Balls that Does Not Cover R

Despite our ability to place open balls around each point in Q without encompassing R, we can achieve this by utilizing balls with shrinking radii, ensuring that their sum remains finite. Let $r_q$ denote the distance from $q$ to $pi$, represented as $|q-pi|$. Consequently, the collection of open balls ${B_{r_q}(q)}_{qin Q}$ will effectively cover all […]

Exploring the Connection Between Topology and Graph Theory

I familiarized myself with the Wikipedia entries on topology, graph theory (including topological graph theory). Is topology solely focused on the examination of shapes, whereas graph theory focuses on relationships, with the convergence occurring in topological graph theory? Question: I have reviewed the Wikipedia pages for both topology and graph theory, including topological graph theory. […]