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