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# Finer Topologies

TOPO-JCSF4J

Suppose $(X,\tau)$ is a topological space. Which of the following are topologies on $X$ that are finer than $\tau$? Checl ALL that apply.

A

$\tau_1=\mathcal{P}(X)$

B

$\tau_2 = \{\emptyset, X\}$

C

Given a point $p\in X$, $\tau_3=\{U\subseteq X : p\in U\}\cup\{\emptyset\}$

D

Given a point $p\in X$, $\tau_4=\{U\subseteq X : p\notin U\}\cup\{X\}$

E

$\tau_5=\{X - U : U\in \tau\}$