Arens fort room

The Arens Fort space , named after the mathematicians RF Arens and MK Fort , is a specially constructed example of a topological space that is often used as a counterexample due to its properties.

definition

Typical zero environment, only columns 2, 3 and 5 do not contain almost all points.

The underlying set is , that is , the set of all pairs of natural numbers . The subset is called the -th column. The set becomes a topological space, the Arens Fort space , by declaring the following sets to be open : ${\displaystyle \mathbb {N} ^{2}}$ ${\displaystyle (m,n)}$ ${\displaystyle m,n=0,1,2,3,4,....}$ ${\displaystyle \{(m,n);n\in \mathbb {N} \}}$ ${\displaystyle m}$ ${\displaystyle \mathbb {N} ^{2}}$

• Any amount in that doesn't include zero . ${\displaystyle \mathbb {N} ^{2}}$ ${\displaystyle (0,0)}$
• Any set that contains the zero point and all but finitely many points in all but finitely many columns.

Topological properties

• The Arens Fort room is a normal Hausdorff room
• Each point is the countable average of closed environments .
• The Arens Fort room is a Lindelöf room
• Exactly the finite subsets are compact .

Missing properties

• The Arens-Fort space does not satisfy either the first or the second countability axiom .
• The Arens Fort room cannot be metrised .
• The Arens Fort room is not compact.

Counterexamples

• In metric spaces, the second axiom of countability follows from the separability . The Arens-Fort space shows that this does not apply in general, because it is separable (it itself only consists of countably many points), but according to the above, it does not satisfy the second axiom of countability.
• If one counts the points as in Cantor's first diagonal argument , one obtains a sequence that always has sequence members in every column and thus in every zero neighborhood. ${\displaystyle \mathbb {N} ^{2}\setminus \{(0,0)\}}$ ${\displaystyle (x_{n})_{n}}$

${\displaystyle {\begin{array}{cccccccccc}x_{6}\\\uparrow &\searrow \\x_{5}&&x_{7}&&\ddots \\&\nwarrow &&\searrow &&\nwarrow \\x_{1}&&x_{4}&&x_{8}&&x_{11}\\&\searrow &&\nwarrow &&\searrow &&\nwarrow \\&&x_{2}&\rightarrow &x_{3}&&x_{9}&\rightarrow &x_{10}\end{array}}}$

${\displaystyle (0,0)}$is the only accumulation point of this sequence, but no subsequence of this sequence converges to . ${\displaystyle (0,0)}$

• Subspaces of Kelley spaces are generally not Kelley spaces. The Arens-Fort space is not a Kelley space, because the compact subsets are precisely the finite ones, but by means of Stone-Čech compactification it is a subspace of a compact and thus a Kelley space.
• The locally uniform convergence does not follow from the compact convergence . Looking at them through

${\displaystyle f_{k}(m,n)={\begin{cases}1&{\mbox{falls }}m+n{\mbox{ ungerade und }}m+n\leq k\\0,&{\mbox{sonst}}\end{cases}}}$

and

${\displaystyle f(m,n)={\begin{cases}1&{\mbox{falls }}m+n{\mbox{ ungerade}}\\0,&{\mbox{sonst}}\end{cases}}}$

defined functions , the sequence of functions converges point by point to . Since precisely the finite sets are compact, there is even compact convergence. Every function is continuous, because it is constantly equal to 0 in the zero neighborhood , but the limit function is discontinuous because it takes the value 1 in every zero neighborhood. In particular, there is no locally uniform convergence, because otherwise the limit function would have to be continuous. ${\displaystyle \mathbb {N} ^{2}\rightarrow \mathbb {R} }$ ${\displaystyle (f_{k})_{k\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle f_{k}}$ ${\displaystyle \{(0,0)\}\cup (\mathbb {N} ^{2}\setminus \{0,\ldots ,k\}^{2})}$ ${\displaystyle f}$

literature

• Richard Arens : Note of Convergence in Topology. In: Mathematics Magazine . Vol. 23, No. 5, 1950, pp. 229-234, doi: 10.2307 / 3028991 .
• Lynn Arthur Steen , J. Arthur Seebach: Counterexamples in Topology. 2nd edition. Springer, New York NY et al. 1978, ISBN 0-387-90312-7 .