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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.

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    $\begingroup$ Take $\mathbb{R}$. Or any non-compact metric space. $\endgroup$ Nov 5, 2014 at 21:59
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    $\begingroup$ Thanks! I didn't realize that I was staring at the answer for past few hours! - RD $\endgroup$
    – roalddahl
    Nov 5, 2014 at 22:03
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    $\begingroup$ I was going to say, "or any infinite discrete space", but those are subsumed by noncompact metric spaces. All right, take any limit ordinal with the order topology. $\endgroup$
    – bof
    Nov 5, 2014 at 22:04
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    $\begingroup$ Turns out "every normal space is regular" is false, see math.stackexchange.com/questions/1888302/… $\endgroup$ Aug 10, 2016 at 16:17

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$\pi$-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

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