📘 topology

1. The problem involves understanding the union of three open sets $U_1$, $U_2$, and $U_3$ in a topological space $X$ and the concept of faces of a simplex. 2. A face of a simplex

1. **Problem Statement:** Prove that $\mathbb{R}^n \setminus \{0\}$ is simply connected for $n \geq 3$. This means every loop in $\mathbb{R}^n \setminus \{0\}$ can be continuously

1. **Problem statement:** Show that the intersection of two circles is a retract of the torus minus a point. 2. **Understanding the problem:** The torus $T^2$ can be viewed as $S^1

1. **Problem statement:** Show that if $X$ is Hausdorff, then every retract $A$ of $X$ is closed in $X$.

1. **State the problem:** We have a set $X=\{x,y,z,w,t\}$ with topology $\tau=\{X,\emptyset,\{x\},\{x,y\},\{x,z,w\},\{x,y,z,w\},\{x,y,t\}\}$. We need to find the neighborhood syste

1. **Stating the problem:** We need to understand what a $T_1$ space is, give two examples, and then show that a discrete topological space is a $T_3$ space. 2. **Definition of a $

1. مسئله را بیان می‌کنیم: می‌خواهیم مفهوم نقاط داخلی مجموعه‌ها $E$ را در فضاهای متریک مختلف بررسی کنیم. 2. تعریف نقاط داخلی: نقطه‌ای $x$ در مجموعه $E$ داخلی است اگر یک همسایگی باز

1. مسئله را بیان می‌کنیم: دو مثال داریم که در فضای متری (metric space) تعریف شده‌اند و می‌خواهیم مفهوم درونی (interior) مجموعه‌ها را توضیح دهیم. 2. تعریف درونی مجموعه: درونی یک مجم

1. The problem asks whether the parabola $y = x^2$ and the $x$-axis are topologically equivalent to a cup and a donut, and if there are simplicial or sheaf methods in computational

1. **Problem Statement:** Show that $(X, \tau)$ with $X = \{a,b,c\}$ and $\tau = \{\emptyset, X, \{a\}, \{b\}, \{a,b\}\}$ is a topological space and find the closure of the set $\{

1. Let's state the problem: We want to explore an example related to the $T_0$ (Kolmogorov) separation axiom using usual topology, interval topology, or cofinite topology. 2. Recal

1. Topology is a branch of mathematics focused on the properties of space that are preserved under continuous deformations such as stretching and bending, but not tearing or gluing

1. The problem asks for the definition of a neighborhood system. 2. In topology, a neighborhood system (or neighborhood filter) of a point $x$ in a topological space is the collect

1. **Problem:** Prove that for a function $f : X \to Y$ between topological spaces, the following conditions are equivalent: (a) $f$ is continuous; (d) $f(\overline{A}) \subset \ov

1. **Problem statement:** Prove that for a topological space $X$ and subset $A \subseteq X$, the complement of the interior of $A$ equals the interior of the complement of $A$, i.e

1. **Problem Statement:** Show by example that the infinite union of compact sets may not be compact. 2. **Recall the definition:** A set is compact if it is closed and bounded.

1. **Problem Statement:** Prove that the open interval $(a,b)$ is an open subset of the real numbers $\mathbb{R}$. 2. **Definition of an Open Set:** A subset $U$ of $\mathbb{R}$ is

1. **Problem statement:** Given two non-empty subsets $S$ and $T$ of a topological space $(X, \tau)$ such that $S \subseteq T$, show that if $S$ is dense in $X$, then $T$ is also d

1. **Problem Statement:** Given two non-empty subsets $S$ and $T$ of a topological space $(X, \tau)$ such that $S \subseteq T$, and a point $p$ which is a limit point of $S$, we ne

1. **Problem statement:** We are given the set $\mathbb{Z}$ of integers with the finite closed topology and the set $E$ consisting of all even integers. We need to find the limit p

1. **Problem Statement:** We are given the set $(\mathbb{Z}, t)$ where $\mathbb{Z}$ is the set of integers and $t$ is the finite closed topology. We need to find the limit points o