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 […]

# Category: General topology

## Criteria for the continuity of a function

For a function to be continuous , the epsilon delta definition of continuity simply needs to hold, so there are no breaks or holes in the function (in the 2-d case). For a function to be smooth, it has to have continuous derivatives up to a certain order, say k. We say that function is […]

## Can the ArzelĂ â€“Ascoli theorem be adapted to apply to $C([0,infty))$?

The inequality states that the supremum of the absolute difference between the values of a function at two points within a certain distance of each other is less than or equal to one for all functions in F. By setting delta equal to half of delta 1, which is less than delta 1, we can […]

## 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 $gamma_1$, […]

## Uncertainty Surrounding Munkres’ Explanation of Topology Basis

The way to determine the topology generated by a basis is as follows: A subset $mathcal O$ of $X$ is considered open if there exists an element $mathcal B$ in the basis such that $xin mathcal Bsubseteq mathcal O$ for all $xin mathcal O$. Alternatively, open subsets are defined as unions of basis elements. For […]

## 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 […]

## Comprehending the Interior of a Set A through Basic Topology

In other words, if $x$ belongs to any open set within $A$, it must also belong to the interior of $A$. This can be stated as a definition. Another way to define closed sets is by taking their complements with respect to open sets. Question: In my writing, I offer two distinct definitions, which, when […]

## 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. […]