Problem 1: Prove that for every non-empty set $X$ there is a topology on $X$ with only 2 elements.
Problem 2: Find all of the topologies that you could make using the set \(X = \{a, b, c\}\).
Problem 3: Is it possible for a topology to only have 1 element?
Problem 4: Is the set of all subsets of a set $X$ a topology on $X$?
Problem 5: Give an example of a set $X$ and a topology $\mathcal{T}$ on $X$ and a subset $T_0$ of $\mathcal{T}$ such that the intersection of all elements in $T_0$ is not in $\mathcal{T}$.
Problem 6: If $X$ has $n$ elements and $k$ is an integer greater than 1 and less than $n+2$, show that you could find a topology on $X$ with $k$ elements.
Problem 7: Let $X$ be a set with $n$ elements. Up to how many elements could a topology on $X$ have?
Problem 8: Is it possible for a topology $\mathcal{T}$ on a set $X$ to have an infinite number of elements but every element of $\mathcal{T}$ except $X$ has a finite number of elements?
Problem 9: Let $X$ be a set and suppose $\mathcal{T}$ is a set of subsets of $X$ such that for any two elements of $\mathcal{T}$, one of them is a subset of the other. Also, assume that the empty set and $X$ are in $\mathcal{T}$. Is $\mathcal{T}$ necessarily a topology on $X$?
Problem 10: If $(X, \mathcal{T})$ is a topological space and $X \subseteq Y$, then prove that $(Y, \mathcal{T} \cup \{Y\})$ is also a topological space.
Problem 11: If $\mathcal{T}$ is a topology on $X$ and $\mathcal{T}$ has an infinite number of elements, show that $X$ has an infinite number of elements.
Problem 12: Prove that a finite union of closed sets is closed and that an arbitrary intersection of closed sets is closed.
Problem 13: Let $X$ be a set and let $\mathcal{T} = \{U \subseteq X : |X \setminus U | \in \mathbb{Z} \}$. Show that $\mathcal{T}$ is a topology on $X$.