Revisions to When are all continuous self-maps of a topological spaces generated by retractions and self-homeomorphisms of prime order?

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edited Sep 12, 2013 at 9:41

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This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true?

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.

Edit (after reading some of the answers):

Does anyone know the answer for Stone spaces? Are there any examples of Stone spaces, >where the answer to the question is no?

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true?

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.

Edit (after reading some of the answers):

Does anyone know the answer for Stone spaces? Are there any examples of Stone spaces, >where the answer to the question is no?

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true?

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.

added 191 characters in body

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edited Sep 9, 2013 at 8:48

Niemi

1.5k
14
23

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true?

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.

Edit (after reading some of the answers):

Does anyone know the answer for Stone spaces? Are there any examples of Stone spaces, >where the answer to the question is no?

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true?

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true?

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.

Edit (after reading some of the answers):

Does anyone know the answer for Stone spaces? Are there any examples of Stone spaces, >where the answer to the question is no?

deleted 166 characters in body

Source Link

edited Sep 6, 2013 at 15:49

Niemi

1.5k
14
23

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true? All I know is that it is true if the topology is discrete or trivial (so we are essentially talking about sets and functions), although is think that it requires AC.

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true? All I know is that it is true if the topology is discrete or trivial (so we are essentially talking about sets and functions), although is think that it requires AC.

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.

This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

the self-homeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a characterization of the cases where it is true?

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.