Topological Spaces

due by Wednesday, Aug 20, 2025

1. Let $X=\{1, 2\}$. What is the number of collections of subsets $\Omega\subset 2^X$ that form a topology on $X$? Prove your answer.
2. Let $X$ be a set, $\Omega$ a topology on $X$ and $Y$ be the set obtained from $X$ by adding a single point $\infty$ to $X$. Show that $$\{\{\infty\}\cup U: U \in \Omega\}\cup\{\phi\}$$ is a topology on $Y$.