Let $X=\{a,b,c,d\}$. Find the connected component $C(a)=\bigcup \{A \mid a \in A , A \subset X, \text{$A$ connected}\}$.

Let $X=\{a,b,c,d\}$ and $\tau=\{\emptyset, \{a,b\}, \{a,b,c\}, \{c,d\}, X\}$. Find the connected component $C(a)=\bigcup \{A \mid a \in A , A \subset X, \text{$A$ connected}\}$.

I'm trying to build intuition for connected components of topological space and I have trouble with determining if a subset of a space $X$ is connected.

I know that a space is connected if for some non-empty $U,V \in \tau$ we have that $U \cap V =X$ and $U \cap V = \emptyset$.

In this case I think I should consider sets $\{a\}, \{a,b\}, \{a,c\}, \{a,d\}, \{a,b,c\}, \{a,c,d\}, \{a,b,d\}, \{a,b,c,d\}$. I think these cover all the subsets of $X$ with $a$ fixed since there is $2^{n-1}$ subsets of a set with one element fixed.

But I don't know how to check whether these sets are connected? A subspace seems to be connected if it's connected in it's relative topology, but what is the relative topology of these subsets anyway here?

Are the relative topologies here $$\tau_{\{a\}}=\{\emptyset, \{a\}\} \\\tau_{\{a,b\}}=\{\emptyset, \{a,b\}\} \\\tau_{\{a,c\}}=\{\emptyset,\{a\}, \{a,c\}, \{c\}\}\\ \tau_{\{a,d\}}=\{\emptyset, \{a\}, \{a,c\}, \{c\}\} \\ \tau_{\{a,b,c\}}=\{\emptyset, \{a,b\},\{a,b,c\}, \{c\}\} \\ \tau_{\{a,c,d\}}=\{\emptyset, \{a\},\{a,c\},\{c,d\}\} \\ \tau_{\{a,b,d\}} = \{\emptyset,\{a,b\},\{d\}\}$$ and for the last set $\{a,b,c,d\}$ the topology would just be $\tau$ as it's the universal set.

"I know that a space is connected if for some non-empty $U,V \in \tau$ we have that $U \cap V =X$ and $U \cap V = \emptyset$."

Attention: there's a mistake, it should be: $U,V \in \tau$ we have that $U \cup V =X$ and $U \cap V = \emptyset$

And in order to find the conntected component $C(a)$ you must be given a topology $\tau$ What's the topology on X in your exercise?