Consider a set $E$ with some point $p \in E$ and the topology: $T = \left \{ \emptyset \right \}\cup \left \{ C \subset E: p \in E \right \}$. Is this topology, separable, first countable or second countable?
I feel like it depends on the size of $E$, because $|P(E)| = 2^{|E|}$ but I'm struggling to figure out what a base for this topology would look like which makes it difficult to answer the questions about countability. Any solutions or hints would be appreciated!
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Okay, so I'm still not sure about countability but the set $\left \{ p \right \}$ is in fact dense because the only closed set containing $p$ is $E$ itself meaning that the closure $\bar{\left \{ p \right \}}$ is the intersection of only one set $E$ meaning that the set is dense and because the set is countable (finite in fact), the space is separable.
This is knwon as the particular point tpology. You can find all you need in the wikipedia page. By the way $p$ is always dense so $T$ is always separable, for any $x$ the set $\{x,p\}$ is a local basis at $x$, so $T$ is first countable. As you mention, the second countability depends on the cardinality of $E$.