Let us have a look at the following statement: every pathwise connected Hausdorff space is arcwise connected. It holds true in ZFC or even in ZF+an appropriate weaker form of AC. I wonder if someone knows whether there is a model for ZF in which a pathwise connected Hausdorff space can be not arcwise connected. I would appreciate correct answers.
This is just what I suspect. Very few scientists answer some of my questions. Most do not answer my questions at all because they do not know any reasonable answer to the questions. So, the lack of their answer is the answer that they simply do not know and, even worse, they may not want to know.
Dear Mani A.,
Thank you for your suggestion. I've thoght that people working on connected spaces should be interested in my question, too. Perhaps, many of them have not noticed my question. I shall think about a correct answer in my spare time.
Eliza, please keep on posting your questions, they are very inspiring for those who, like myself, are interested in the research of what happens in the absence of the axiom of choice.
Dear Samuel Gomes da Silva,
Thank you for your encouragenent. Have you noticed that the axiom of choice is equivalent in ZF to the following statement: For every non-void set $I$ and for every collection $\{ X_i; i\in I\}$ of non-void topological spaces, if the product $\prod_{i\in I}X_i$ is compact, then all the spaces $X_i$ are also compact where $i\in I$.? Of course, AC is also equivalent to the statement: For every non-void set $I$ and for every collection $\{ X_i; i\in I\}$ of non-void topological spaces, if the product $\prod_{i\in I}X_i$ is connected, then all the spaces $X_i$ are also connected where $i\in I$. Many other equivalents of AC in topology that are not mentioned in Herrlich's excellent book can be found. Eliza
Yes, indeed I have published (together with a former student) an equivalence of the Axiom of Choice which is similar to those you refer to: “If a product of a non-empty family of sets is closed in a topological (Tychonoff) product, then at least one of the factors is closed”. Acta Mathematica Hungarica, October 2011, Volume 133, Issue 1, pp 128-132. Besides of posting questions here at ResearchGate you may also use sites as MathOverFlow and MathStackExchange, as well as the FOM list which was already suggested above. But I do encourage you to keep on posting your questions here, they are very interesting. Thank you.
Dear Samule Gomes da Silva,
Thank you for your response very much. I shall look at your article printed in Acta Math. Hung. in 2011. I hope to be able to complete my own article soon. I shall submit its draft to RG.
Eliza
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