On Tychonoff's type theorem via grills

Document Type : Research Paper


Department of Mathematics‎, ‎Payam Noor University‎, ‎Tehran‎, Iran.


‎Let $\{X_{\alpha}:\alpha\in\Lambda\}$ be a collection of topological spaces‎, ‎and $\mathcal {G}_{\alpha}$ be a grill on $X_{\alpha}$ for each $\alpha\in\Lambda$‎. ‎We consider Tychonoff\rq{}s type Theorem for $X=\prod_{\alpha\in\Lambda}X_{\alpha}$ via the above grills and a natural grill on $X$ ‎related to these grills, and present a simple proof to this theorem‎. ‎This immediately yields the classical theorem of Tychonoff. We shall also observe> that the above result is also equivalent to the Axiom of Choice‎.‎


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