Articles of general topology

Question about Definition of homeomorphism (counter example)

I’m teaching my self topology with the aid of a book, but i’m confused about the meaning of homeomorphic. Below, I have 2 topologies, $\mathscr{T}_1$ and $\mathscr{T}_2$ and I’m pretty sure they are not homeomorphic, but I can make a continuous bijection between them. Definition of homeomorphism: If $X$ and $Y$ are topological spaces, a […]

Sum of closed convex set and unit ball in normed space

Let $(X, \|.\|)$be a real normed space. Let $A$ be a closed convex suset of $X$ and $\mathbb{B}$ a unit ball in X, i.e. $$ \mathbb{B}=\{x\in X: \|x\|\leq 1\}. $$ I would like to ask whether $A+\mathbb{B}$ is still closed.

Uniqueness of continuous extension from $A$ to $\overline{A}$ for maps into a Hausdorff space

I want to prove the following. Let $A$ be a subset of $X$. Let $f:A \to Y$ be continuous. Let $Y$ be Hausdorff. Show that if $f$ can be extended to a continuous function $g:\overline{A}\to Y$, then $g$ is uniquely determined by $f$. Assume the contrary.$g_{1}$, and $g_{2}$ are the extended $f$. Consider $B=\lbrace x\in\bar{A}\vert […]

Simply connectedness in $R^3$ with a spherical hole?

I understand why $R^3 – {(0,0,0)}$ is simply connected, and I also understand why $R^2 – {(0,0)}$ is not simply connected. The way I look it at is if checking if the region is $a)$ path-connected and $b)$ any curve can be contracted to a point in the region. From what I reasoned it seems […]

Fundamental group of multiplicative group in Zariski topology

What is the fundamental group of the multiplicative group of the complex numbers $\mathbb{G}_m(\mathbb{C})$ with respect to the Zariski topology. More precisely, what are the homotopy classes of continuous loops $f:[0,1]\rightarrow \mathbb{G}_m(\mathbb{C})$ with a fixed base point?

Need help with understanding the Basis for a Topology

I don’t have a particular problem to share but I’m asking for assistance in describing a Basis or Sub-Basis for some Topology T. So, if I have some set, say $\{a,b,c\}$ or $\{a,b\}$ what will the basis and sub-basis be? I understand the definition of a Topology and how to find a topology for a […]

show that $f$ is continuous

Show that $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous in the $\delta-\epsilon$ definition of continuity if and only if for all $x \in \mathbb{R}$ and all open set $U$ where $f(x) \in U$, there exists an open set $V$ where $x \in V$ and $f(V) \subset U$ Forward direction: Let open set $V$ such that $V \subset […]

Finite mapping $f : \mathbb R^2 \to \mathbb R$

Is there an continous function $f: \mathbb R^2 \to \mathbb R$ such that $f^{-1}(a)$ is finite for every $a \in \mathbb R$? It’s not possible for analytic or smooth but I’m curious about continous mapping.

A universal property for the subspace topology

Let $X$ be topological space and $Y$ be a subset of $X$ with $i\colon Y\to X$ the inclusion map. Show that the induced topology of $Y$ is characterized by the following property: A function $f\colon Z \to Y$ of a topological space $Z$ into $Y$ is continuous if and only if $i\circ f$ is continuous.

Finding limit points in lexicographic order topology

If we consider the unit square (i.e $[0,1] \times [0,1]$) with the lexicographic order induced by ${\mathbb{R}}^{2}$, what are the limit points of the unit square? My answer is: the set of limit points is equal to: $ [0,1]^{2} \setminus ((0,1) \times \{1\}))$. Is my answer correct? If not, how do you find them? I […]