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.


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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

1 Answer 1


$\pi$-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 $\mathbb{R}^\omega$


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


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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