Read More
Date: 23-5-2021
![]()
Date: 15-8-2021
![]()
Date: 22-9-2016
![]() |
The topology on the Cartesian product of two topological spaces whose open sets are the unions of subsets
, where
and
are open subsets of
and
, respectively.
This definition extends in a natural way to the Cartesian product of any finite number of topological spaces. The product topology of
![]() |
where is the real line with the Euclidean topology, coincides with the Euclidean topology of the Euclidean space
.
In the definition of product topology of , where
is any set, the open sets are the unions of subsets
, where
is an open subset of
with the additional condition that
for all but finitely many indices
(this is automatically fulfilled if
is a finite set). The reason for this choice of open sets is that these are the least needed to make the projection onto the
th factor
continuous for all indices
. Admitting all products of open sets would give rise to a larger topology (strictly larger if
is infinite), called the box topology.
The product topology is also called Tychonoff topology, but this should not cause any confusion with the notion of Tychonoff space, which has a completely different meaning.
REFERENCES:
Cullen, H. F. Introduction to General Topology. Boston, MA: Heath, pp. 65-91, 1968.
Joshi, K. D. "Product Topology." §8.2 in Introduction to General Topology. New Delhi, India: Wiley, pp. 196-203, 1983.
McCarty, G. "Tychonoff for Two." In Topology, an Introduction with Application to Topological Groups. New York: McGraw-Hill, pp. 154-157, 1967.
Willard, S. "Product Spaces, Weak Topologies." §8 in General Topology. Reading, MA: Addison-Wesley, pp. 52-59, 1970.
|
|
النوم 7 ساعات ليلا يساعد في الوقاية من نزلات البرد
|
|
|
|
|
اكتشاف مذهل.. ثقب أسود ضخم بحجم 36 مليار شمس
|
|
|
|
|
مركز ويلسون الأمريكي ينشر مقالًا للمركز العراقي لتوثيق جرائم التطرف عن أهمية أرشفة جرائم البعث
|
|
|