Abel's identity

In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum. R.H. Bing proved that, in a certain well-defined sense, most continua in Rⁿ, 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 descovered the first example of a homogeneous hereditarily indecomposable continuum K, later named the pseudo-arc, giving a negative answer to the 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.

Construction

The following construction of the pseudo-arc follows Template:Harv.

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 $\mathcal{𝒞}=\{C_1,C_2,\dots ,C_n\}$ in a metric space such that $C_i\cap C_j\ne \mathrm{\varnothing }$ if and only if $|i-j|\le 1.$ 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 mth link of the larger chain to the nth, the smaller chain must first move in a crooked manner from the mth link to the (n-1)th link, then in a crooked manner to the (m+1)th link, and then finally to the nth link.

More formally:

Let $\mathcal{𝒞}$ and $\mathcal{𝒟}$ be chains such that

1. each link of $\mathcal{𝒟}$ is a subset of a link of $\mathcal{𝒞}$, and
2. for any indices i, j, m, and n with $D_i\cap C_m\ne \mathrm{\varnothing }$, $D_j\cap C_n\ne \mathrm{\varnothing }$, and $m<n-2$, there exist indices $k$ and $\ell$ with $i<k<\ell <j$ (or $i>k>\ell >j$) and $D_k\subseteq C_{n-1}$ and $D_{\ell }\subseteq C_{m+1}.$

Then $\mathcal{𝒟}$ is crooked in $\mathcal{𝒞}.$

Pseudo-arc

For any collection C of sets, let $C^{*}$ denote the union of all of the elements of C. That is, let

$C^{*}=\bigcup _{S\in C}S.$

The pseudo-arc is defined as follows:

Let p and q be distinct points in the plane and ${\left\{{\mathcal{𝒞}}^i\right\}}_{i\in \mathbb{\mathbb{N}}}$ be a sequence of chains in the plane such that for each i,

1. the first link of ${\mathcal{𝒞}}^i$ contains p and the last link contains q,
2. the chain ${\mathcal{𝒞}}^i$ is a $1/2^i$-chain,
3. the closure of each link of ${\mathcal{𝒞}}^{i+1}$ is a subset of some link of ${\mathcal{𝒞}}^i$, and
4. the chain ${\mathcal{𝒞}}^{i+1}$ is crooked in ${\mathcal{𝒞}}^i$.

Let

$P=\bigcap _{i\in \mathbb{\mathbb{N}}}{\left({\mathcal{𝒞}}^i\right)}^{*}.$

Then P is a pseudo-arc.

References

• R.H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J., 15:3 (1948), 729–742
• R.H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math., 1 (1951), 43–51
• George W. Henderson, Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc. Annals of Math., 72 (1960), 421–428
• Bronisław Knaster, Un continu dont tout sous-continu est indécomposable. Fundamenta math. 3 (1922), 247–286
• Wayne Lewis, The Pseudo-Arc, Bol. Soc. Mat. Mexicana, 5 (1999), 25–77
• Edwin Moise, An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc., 63, no. 3 (1948), 581–594
• Sam B. Nadler, Jr, Continuum theory. An introduction. Pure and Applied Mathematics, Marcel Dekker (1992) ISBN 0-8247-8659-9