📘 topology

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

1. Let's clarify the problem: We want to understand why the set difference $B \setminus C$ is closed, given that $B$ and $C$ are subsets of a topological space. 2. Recall that $B \