📘 topology

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

1. **Problem Statement:** We are given the set $A = \{1, 2, 3, \ldots, 10\}$ as a subset of the integers $\mathbb{Z}$ equipped with the finite closed topology. We need to find the

1. Problem: Find the limit points of given sets in the finite closed topology on integers $(\mathbb{Z}, T)$. (i) Set $A = \{1, 2, 3, \ldots, 10\}$.

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 have a sequence of closed intervals $F_i = [a_i, b_i]$ with $F_1 \subseteq F_2 \subseteq F_3 \subseteq \cdots$. We want to provide examples showing tha

1. **Problem statement:** We want to find a closed set $E \subseteq \mathbb{R}^2$ such that the projection onto the second coordinate $\pi_2(E)$ is closed, but the projection onto

1. **Problem statement:** (a) Show that the set $E = \{(x, \frac{1}{x}) \mid x > 0\}$ is closed in $\mathbb{R}^2$ but its projection $\pi_1(E)$ is not closed.

1. **Problem 12:** Identify \( \operatorname{cl}_{A \cup B}(B) \) where \( A = \{ z \in \mathbb{C} : |z + 1|^2 \leq 1 \} \) and \( B = \{ z \in \mathbb{C} : |z - 1|^2 < 1 \} \). 2.

1. The problem is to understand what it means for a set $C$ to be neither open nor closed in a topological or metric space. 2. A set is **open** if it contains none of its boundary