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In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology.The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that ...
The product of any collection of compact spaces is compact. (This is Tychonoff's theorem , which is equivalent to the axiom of choice .) In a metrizable space , a subset is compact if and only if it is sequentially compact (assuming countable choice )
The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that requires the axiom of choice and is equivalent to it in its most general formulation, [3] and shows why the product topology may be considered the more useful ...
Tube Lemma — Let and be topological spaces with compact, and consider the product space. If N {\displaystyle N} is an open set containing a slice in X × Y , {\displaystyle X\times Y,} then there exists a tube in X × Y {\displaystyle X\times Y} containing this slice and contained in N . {\displaystyle N.}
As a product of compact Hausdorff spaces, the Hilbert cube is itself a compact Hausdorff space as a result of the Tychonoff theorem. The compactness of the Hilbert cube can also be proved without the axiom of choice by constructing a continuous function from the usual Cantor set onto the Hilbert cube.
Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called Tychonoff's theorem, is yet another equivalence to the axiom of choice.
The product of a finite number of σ-compact spaces is σ-compact. However the product of an infinite number of σ-compact spaces may fail to be σ-compact. [11] A σ-compact space X is second category (respectively Baire) if and only if the set of points at which is X is locally compact is nonempty (respectively dense) in X. [12]
A topological space is called a Tychonoff space (alternatively: T 3½ space, or T π space, or completely T 3 space) if it is a completely regular Hausdorff space. Remark. Completely regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence.