Real line $\mathbb R^1$ with the countable complement topology is not compact
47251
Non-compactness of $\mathbb{R}$ with the cocountable topology
81927
Example of closed and bounded in $\mathbb Q$ set that is not compact
95635
$\ell^{\infty}(\mathbb N)$ is not a separable space
102485
Showing that a topological space is ${\rm T}_1$
131637
Real projective space is Hausdorff: is this proof correct?
7678
Show "countable complement topology" is a topology
62793
$\mathbb{R}$ with the lower limit topology is not second-countable
36631
Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
80735
Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact
70413
A vector space over $R$ is not a countable union of proper subspaces
120188
Why is the real line not used in Descriptive Set Theory?
131082
Proof that $\mathbb R/[0,1]$ and $\mathbb R$ are homeomorphic
23837
Why is $\mathbb{R}/{\sim}$ not first countable at $[0]$, where $x \sim y \Leftrightarrow x = y\text{ or }x,y \in \mathbb{Z}$?
95321
Subspaces of all real-valued continuous functions on $\mathbb{R}^1$
41872
Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$
47231
The space $C^0([0;1],\mathbb{R})$ of all continuous, real-valued functions on $[0,1]$ is not reflexive.
32819
Looking for examples of first countable, compact spaces which is not separable