Alexandroff plank explained

Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example.

Definition

The construction of the Alexandroff plank starts by defining the topological space

(X,\tau) to be the Cartesian product of [ 0 ,\omega _{1]} and [ -1 , 1 ], where \omega _{1} is the first uncountable ordinal, and both carry the interval topology. The topology \tau is extended to a topology \sigma by adding the sets of the form$$U(\backslash alpha,n)\; =\; \backslash \; \backslash cup\; (\backslash alpha,\backslash omega\_1]\; \backslash times\; (0,1/n)$$where p = (\omega _{1,0)} \in X.

The Alexandroff plank is the topological space

(X,\sigma).

It is called plank for being constructed from a subspace of the product of two spaces.

Properties

The space

(X,\sigma) has the following properties:

1. It is Urysohn, since

(X,\tau) is regular. The space (X,\sigma) is not regular, since C = \{(\alpha, 0 ) : \alpha < \omega _{1\}} is a closed set not containing (\omega _{1,0),} while every neighbourhood of C intersects every neighbourhood of (\omega _{1,0).}

1. It is semiregular, since each basis rectangle in the topology

\tau is a regular open set and so are the sets U(\alpha,n) defined above with which the topology was expanded.

1. It is not countably compact, since the set

\{(\omega _{1,-1/n)} : n =2 , 3 , ... \} has no upper limit point.

1. It is not metacompact, since if

\{V _{\alpha\}} is a covering of the ordinal space [ 0 ,\omega _{1)} with not point-finite refinement, then the covering \{U _{\alpha\}} of X defined by U _{1} = \{( 0 ,\omega _{1)\}} \cup ([ 0 ,\omega _{1]} x ( 0 , 1 ]), U _{2} = [ 0 ,\omega _{1]} x [ -1 , 0 ), and U _{\alpha} = V _{\alpha} x [ -1 , 1 ] has not point-finite refinement.

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).
• S. Watson, The Construction of Topological Spaces. Recent Progress in General Topology, Elsevier, 1992.