Is there a non-compact Hausdorff space $Y $ such that for every topological space X, every function $f:X\to Y$ with closed graph is continuous?

Question

Closed Graph Theorem (Topological Version , necessary condition ) : $X,Y$ be two topological space where $ Y $ is a compact Hausdorff space and $f:X\to Y$ be a map with $G_f=\{(x,f(x)):x\in X\}\subset X×Y$ is closed in the product topology on $X×Y$ then $f$ is continuous.

Here, I want to replace the target space $Y$ by:

1. a Locally Compact Hausdorff space.

2. a Lindelöf space.

3. a Countably Compact space.

4. a Regular space.

5. a Normal space.

6. a Baire space.

Question : For which of the above space/spaces the conclusion of the closed graph theorem is still true?

$f:\Bbb{R}\to \Bbb{ R}$ defined by

$$f(x)=\begin{cases}\frac1x\quad\text{if}\quad x\ne0\\ 0 \quad\text {otherwise } \end{cases}$$ is discontinuous function with closed graph.

Here, $\Bbb{R}$ is locally compact Hausdorff space, Lindelöf, countably compact, regular, normal, Baire space.

Question: Does any weak form of compactness can force the conclusion of the closed graph theorem to be true?

Question Is there a non-compact Hausdorff space $Y $ such that for every topological space X, every function $f:X\to Y$ with closed graph is continuous?

The target space $Y$ can't have the indiscrete topology.