Am I correct in saying that $P(\{a\}) = \{\{a\}, \emptyset\}$, $P(P(\{a\})) = \{\{\{a\}\}, \{a\}, \emptyset, \{\{a\},\emptyset\}\}$? I'm getting a bit confused over the elements of repeated power sets of a singleton set. 
Could someone clarify the for me P({a}) through P(P(P(P({a})))) ? 
Thanks!
$$P(\{a\})=\{\{a\},\{\}\}$$
 $$P(P(\{a\}))= \{\{a\},\{\}\ , \{\{a\}\} ,\{\{a\},\{\}\} \} etc?$$

$$\mathcal{P}(\{a\})=\{ \varnothing, \{a\}\}$$
$$\mathcal{P(\mathcal{P}(\{a\}))}=\mathcal{P}(\{ \varnothing, \{a\}\})=\{ \varnothing, \{ \varnothing, \{a\}\}, \{ \varnothing\}, \{ \{a \}\}\}$$
PS:If $A=\{a,b\}$ is a set, then:
$$\varnothing \in \mathcal{P}(A), \{a,b\} \in \mathcal{P}(A), \{a\}, \{b\} \in \mathcal{P}(A)$$
