I mainly wanted to check if my solution was correct. (a)-(c) I pretty much have, it is mainly (d) that I am worried about:
Question: Let $(\Omega, \tau)$ be a finite topological space.
a) Show for any $x \in \Omega$ there exists a smallest open set containing $x$, denoted $U_x$.
b) Prove $U_x$ is both connected and compact.
c) Prove $U_x$ is also path connected.
d) Prove that $U_x$ is homotopically equivalent to a point.
e) Determine $\pi_1(U_x,x)$
Attempted Solution:
(a) Because $\Omega$ is finite, let $\tau=\{U_1,...,U_n\}$ be an ordering of the topology such that $|U_i| \leq |U_{i+1}|$. Let $i\leq n$ be the first positive integer such that $x \in U_i$ and let $U_x=U_i$.
(b) All finite topological spaces are compact, hence $U_x$ is compact. To see connectedness, suppose not, then we have non-empty open sets $U,V$ such that $U \cup V = U_x$ but $U \cap V=\emptyset$. WLOG say $x \in U$, then $U_x$ was not the smallest open set containing $x$, a contradiction.
(c) A finite topological space is connected iff it is path connected.
(d) If $U_x$ is already a singleton, we are done. Otherwise define the homotopy as follows: $F(U_x, 0) = x$ but $F(U_x,t)=\textit{id}_{U_x}$
(e) By (d) we clearly have $\pi_1(U_x,x)=1$ the trivial group.