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Igor Belegradek
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I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

UPDATE:

  1. First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).

  2. The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset</> is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather, people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets": any space with this property is countable! Thus we reach a sad conclusion: looking asat closed subsets of infinite deficiency does not give any new examples of spaces that always separate the Hilbert cube no matter how we embed them.

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

UPDATE:

  1. First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).

  2. The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset</> is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather, people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets": any space with this property is countable! Thus we reach a sad conclusion: looking as closed subsets of infinite deficiency does not give any new examples of spaces that always separate the Hilbert cube.

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

UPDATE:

  1. First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).

  2. The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset</> is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather, people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets": any space with this property is countable! Thus we reach a sad conclusion: looking at closed subsets of infinite deficiency does not give any new examples of spaces that always separate the Hilbert cube no matter how we embed them.

update
Source Link
Igor Belegradek
  • 29.1k
  • 2
  • 80
  • 176

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

UPDATE:

  1. First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).

  2. The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset</> is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather, people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets": any space with this property is countable! "Absolute $Z$-sets"Thus we reach a sad conclusion: looking as closed subsets of infinite deficiency does not give any new examples of spaces that always separate the Hilbert cube.

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

UPDATE:

  1. First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).

  2. The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset</> is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets".

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

UPDATE:

  1. First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).

  2. The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset</> is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather, people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets": any space with this property is countable! Thus we reach a sad conclusion: looking as closed subsets of infinite deficiency does not give any new examples of spaces that always separate the Hilbert cube.

update
Source Link
Igor Belegradek
  • 29.1k
  • 2
  • 80
  • 176

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

UPDATE:

  1. First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).

  2. The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset</> is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets".

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?

UPDATE:

  1. First of all, a correction: the above mentioned Lemma 2.1 needs an assumption that the image of the embedding is closed. Without this assumption I can only show that the complement of a finite dimensional subset of a Hilbert cube is connected. (The proof is the same where instead of Alexander duality one uses the result that codimension two subspaces of $\mathbb R^n$ have connected complements).

  2. The simplest example of a subset of the Hilbert cube with path-connected complement is that of deficiency $\ge 2$, where deficiency of a subset</> is the number of coordinate projections mapping the subset to a point. I could not find any study of subsets of deficiency $\ge 2$. Rather people studied closed subsets of infinite deficiency, or their images under ambient homeomorphisms which are also known as the $Z$-sets. So one could ask for conditions on a space such that any embedding in the Hilbert cube is a $Z$-set. The definitive answer is given be Lenaburg in "Absolute $Z$-sets".

Source Link
Igor Belegradek
  • 29.1k
  • 2
  • 80
  • 176
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