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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.
$\pi$-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
