Example of space that is normal but not compact We know that every compact Hausdorff space is normal. Also, every normal space is regular, and hence Hausdorff. However, does normality implies compactness? I think it does not but I can't think of a counterexample: a normal space that is not compact. Could someone please enlighten? Any help will be greatly appreciated.
RD.
 A: $\pi$-Base is an online database inspired by Steen and Seebach's Counterexamples in Topology. It returns forty-eight normal spaces that are not compact. You can visit the search result to read more about any of these spaces.
Appert Space
Arens-Fort Space
Baire Product Metric on $\mathbb{R}^\omega$
Baire space
Bing's Discrete Extension Space
Boolean Product Topology on $\mathbb{R}^\omega$
$C[0,1]$
Cantor's Leaky Tent
Cantor's Teepee
Countable Discrete Topology
Deleted Integer Topology
Discrete Rational Extension of $\mathbb{R}$
Divisor Topology
Duncan's Space
Euclidean Topology
Evenly Spaced Integer Topology
Fortissimo Space
Hilbert Space
Hjalmar Ekdal Topology
Lusin Set
$\mathbb{Z}^\mathbb{Z}$
Metrizable Tangent Disc Topology
Michael's Closed Subspace
Miller's Biconnected Set
Nested Angles
Nested Interval Topology
Nested Rectangles
Odd-Even Topology
One-point Lindelofication of $\omega_1$
Open Ordinal Space $[0,\Gamma)$ $(\Gamma < \Omega)$
Open Ordinal Space $[0,\Omega)$
Radial Interval Topology
Right Half-Open Interval Topology
Right Order Topology on $\mathbb{R}$
Rudin's Dowker space
Sierpinski's Metric Space
Single Ultrafilter Topology
The Infinite Broom
The Infinite Cage
The Irrational Numbers
The Long Line
The $p$-adic Topology on $\mathbb{Z}$
The Post Office Metric
The Radial Metric
The Rational Numbers
Topologist's Sine Curve
Uncountable Discrete Topology
Wheel without Its Hub
