Pseudo-arc

In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. Bing proved that, in a certain well-defined sense, most continua in R^''n'', ''n'' ≥ 2, are homeomorphic to the pseudo-arc.

History

In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane R² must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R² that is homeomorphic to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster discovered the first example of a hereditarily indecomposable continuum ''K'', later named the pseudo-arc, giving a negative answer to a Mazurkiewicz question. In 1948, R. H. Bing proved that Knaster's continuum is homogeneous, i.e. for any two of its points there is a homeomorphism taking one to the other. Yet also in 1948, Edwin Moise showed that Knaster's continuum is homeomorphic to each of its non-degenerate subcontinua. Due to its resemblance to the fundamental property of the arc, namely, being homeomorphic to all its nondegenerate subcontinua, Moise called his example ''M'' a pseudo-arc. Bing's construction is a modification of Moise's construction of ''M'', which he had first heard described in a lecture. In 1951, Bing proved that all hereditarily indecomposable arc-like continua are homeomorphic — this implies that Knaster's ''K'', Moise's ''M'', and Bing's ''B'' are all homeomorphic. Bing also proved that the pseudo-arc is typical among the continua in a Euclidean space of dimension at least 2 or an infinite-dimensional separable Hilbert space. Bing and F. Burton Jones constructed a decomposable planar continuum that admits an open map onto the circle, with each point preimage homeomorphic to the pseudo-arc, called the circle of pseudo-arcs. Bing and Jones also showed that it is homogeneous. In 2016 Logan Hoehn and Lex Oversteegen classified all planar homogeneous continua, up to a homeomorphism, as the circle, pseudo-arc and circle of pseudo-arcs. In 2019 Hoehn and Oversteegen showed that the pseudo-arc is topologically the only, other than the arc, hereditarily equivalent planar continuum, thus providing a complete solution to the planar case of Mazurkiewicz's problem from 1921.

Construction

The following construction of the pseudo-arc follows .

Chains

At the heart of the definition of the pseudo-arc is the concept of a ''chain'', which is defined as follows:

:A chain is a finite collection of open sets $\mathcal=\$ in a metric space such that $C_i\cap C_j\ne\emptyset$ if and only if $, i-j, \le1.$ The elements of a chain are called its links, and a chain is called an ε-chain if each of its links has diameter less than ε.

While being the simplest of the type of spaces listed above, the pseudo-arc is actually very complex. The concept of a chain being ''crooked'' (defined below) is what endows the pseudo-arc with its complexity. Informally, it requires a chain to follow a certain recursive zig-zag pattern in another chain. To 'move' from the ''m''th link of the larger chain to the ''n''th, the smaller chain must first move in a crooked manner from the ''m''th link to the (''n''-1)th link, then in a crooked manner to the (''m''+1)th link, and then finally to the ''n''th link.

More formally:

:Let $\mathcal$ and $\mathcal$ be chains such that

:# each link of $\mathcal$ is a subset of a link of $\mathcal$, and
:# for any indices ''i'', ''j'', ''m'', and ''n'' with $D_i\cap C_m\ne\emptyset$, $D_j\cap C_n\ne\emptyset$, and m, there exist indices $k$ and $\ell$ with i (or $i>k>\ell>j$) and $D_k\subseteq C_$ and $D_\ell\subseteq C_.$

:Then $\mathcal$ is crooked in $\mathcal.$

Pseudo-arc

For any collection ''C'' of sets, let $C^$ denote the union of all of the elements of ''C''. That is, let
:$C^*=\bigcup_S.$

The ''pseudo-arc'' is defined as follows:

:Let ''p'' and ''q'' be distinct points in the plane and $\left\_$ be a sequence of chains in the plane such that for each ''i'',

:#the first link of $\mathcal^i$ contains ''p'' and the last link contains ''q'',
:#the chain $\mathcal^i$ is a $1/2^i$-chain,
:#the closure of each link of $\mathcal^$ is a subset of some link of $\mathcal^i$, and
:#the chain $\mathcal^$ is crooked in $\mathcal^i$.

:Let
::$P=\bigcap_\left(\mathcal^i\right)^.$
:Then ''P'' is a pseudo-arc.

Notes

References

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Continuum theory