筑波大学トポロジーセミナー
世話人：川村一宏，平山至大，石井敦，丹下基生，蓮井翔
| 日時 | ２００９年１２月２日（水）１７：００～１８：００ |
|---|---|
| 場所 | 筑波大学 自然系学系Ｄ棟 Ｄ８１４ |
| 講演者 | Dmitri Shakhmatov 氏 （愛媛大学 理工学研究科） |
| 講演題目 | Making a given subset of an abelian group dense in the power of circle |
| アブストラクト | The classical result of Hewitt-Marczewski-Pondiczeri states: If $\tau$ is an infinite cardinal, $I$ is a set such that $|I|\le 2^\tau$, and for every $i\in I$ a space $X_i$ has a dense subset of size $\le\tau$, then the product $X=\prod_{i\in I} X_i$ also has a dense subset of size $\le\tau$. In this lecture we investigate the following ``algebraic version'' of this theorem. Let $\kappa$ be an infinite cardinal and $T=R/Z$ be the circle group. Given a fixed subset $S$ of an abelian group $G$, we attempt to find a group homomorphism $\pi:G\to T^\kappa$ such that $\pi(S)$ becomes dense in $T^\kappa$. Of particular interest is the special case when $\pi$ can be chosen to be a monomorphism, that is, when the group $G$ and the subgroup $\pi(G)$ of $T^\kappa$ become isomorphic. We will completely resolve this problem in our talk. |
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