# product topology

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## product topology

[′prä‚dəkt tə′päl·ə·jē]
(mathematics)
A topology on a product of topological spaces whose open sets are constructed from cartesian products of open sets from the individual spaces.
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We cover {[A.sub.1]} x [S.sub.1] by the basis elements {U x V}(for the g[ALEPH] product topology) lying in N.Since {[A.sub.1]} x [S.sub.2] is g[ALEPH] compact,{U x V} has a finite subcover, say a finite number of basis elements [U.sub.1] x [V.sub.1],...,[U.sub.n] x [V.sub.n].
Let us remind that, when A and C are topological algebras, then A x C = {(a, c): a [member of] A, c [member of] C}, equipped with the product topology, is also a topological algebra with respect to the algebraic operations defined by
Fortunately, D is still an algebra and, choosing the subspace topology on D, induced by the product topology of A x C, we still obtain a topological algebra and are able to define h : D [right arrow] B by h((a,c)) = f(a) = g(c).
We use the notation from the abstract and A denotes the Cantor set endowed with the standard product topology. Throughout this paper, we also assume that every topological space is of cardinality at least 2.
Let (H, ) be a hypergroupoid and (H, T ) be a topological space, the cartesian product H x H will be equipped with the product topology. The hyperoperation is called:
For any non-empty set [GAMMA] put [SIGMA](T) := {x [member of] [R.sup.[GAMMA]] : {x([gamma]) [not equal to] 0} is countable} endowed with the product topology. It is known that each space [SIGMA]([GAMMA]) is Frechet-Urysohn, see [13].
It is natural to examine whether the topology on the product semigroup is the product topology.
1.[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is Alexandroff (with product topology).
The unitization A x K of A in the product topology is an ([[alpha].sub.n])-galbed algebra if and only if A is an ([[alpha].sub.n])-galbed algebra.
The flow (X, T) is distal if for any x [not equal to] Y the orbit closure of the point (x, y) in X x X (with the product topology) does not meet the diagonal.
Moreover, neutrosophic local compactness and neutrosophic product topology are developed.
(1) T is called a continuous [t.sub.I]-norm if the function T is continuous with respect to the product topology on the set I x I.
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