# Properties of Infinite set on co-finite topology and Countable set on co-countable topology

I am trying to verify some of the properties of infinite set on co-finite topology and countable set on co-countable topology but it is proven to be very tricky because I cannot visualize the spaces. Can someone verify my solution is correct?

1. Infinite set on co-finite topology: $A = (\mathbb{N}, \mathfrak{T}_{co-finite})$
• Open sets: All co-finite sets

• Closed sets: All finite sets

• Separation Axioms (0-4): $A$ is $T_0$ since $\{x\}$ is contained in $\mathbb{N}\backslash\{y\}$, for $x \neq y$.

$A$ is $T_1$ since $\{x\}$ is contained in $\mathbb{N}\backslash\{y\}$, $\{y\}$ is contained in $\mathbb{N}\backslash\{x\}$ for $x \neq y$.

$A$ is not $T_2$ since any open set touches. Therefore not $T_3, T_4$

• Separable: We know that all open set touches in this space, and $\mathbb{N}$ is countable, so there are countably many open sets, so $A$ is separable [something wrong here fixing]

• 2nd Countable (countable base): Not sure...

• 1st Countable (countable local base): Not sure...

1. Countable set on co-countable topology: $B = (\mathbb{N}, \mathfrak{T}_{co-countable})$
• Open sets: All co-countable sets

• Closed sets: All countable sets

• Separation Axioms (0-4): $B$ is $T_0$ since $\{x\}$ is contained in $\mathbb{N}\backslash\{y\}$, for $x \neq y$.

$B$ is $T_1$ since $\{x\}$ is contained in $\mathbb{N}\backslash\{y\}$, $\{y\}$ is contained in $\mathbb{N}\backslash\{x\}$ for $x \neq y$.

I am not sure if $B$ is $T_2$ and above. I only know that uncountable set equipped with co-countable topology is not $T_2$

• Separable: Might not be separable, I am not sure how to prove this.

• 2nd Countable (countable base): Not sure...

• 1st Countable (countable local base): Not sure...

Do these spaces even have bases for us to talk about countability?