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# Strict Coarseness

TOPO-BGRJQ5

Let $X=\{a,b,c\}$ and consider $\tau = \{U\subseteq X : b\in U\}$. $\tau$ defines a topology on $X$. Which of the following topologies on $X$ are strictly coarser than $\tau$?

A

$\tau_1=\{U\subseteq X : a\in U\}$

B

$\tau_2 = \mathcal{P}(X)$

C

$\tau_3=\{\emptyset, \{b\}, \{c\}, \{b,c\}, X\}$

D

$\tau_4=\{\emptyset,\{b\}, \{a,b\}, \{b,c\}, X\}$

E

$\tau_5=\{U\subseteq X : b,c\in U\}$