Connected and Path Connected in Topological Space

Question

Let $X = \{a, b, c\}$ with the topology $\{∅, \{a\}, \{b, c\}, \{a, b, c\}\}$.

Is $X$ connected?

Is $X$ path connected?

Answer 1

Here $X= \{ a \} \cup \{ b,c \}$. So $X$ is not connected.

Answer 2

Connected:

We say not connected if we can separate $X$ i.e. partition $X$ into open sets.

We can always partition any $X$ like

$$X = \{a,b\} \cup \{c\}$$

But the issue is if we can partition into open sets. The above partition is not a separation of $X$ because the sets are not open. However,

$$X = \{a\} \cup \{b,c\}$$

Path connected:

Path connected sets are connected (proofwiki has a proof. For a proof without using that $[0,1]$ is connected, it likely doesn't exist). Therefore, $X$ is not path connected.