Product of topological spaces

A product of topological spaces is a topological space obtained as a set by the Cartesian product of the original topological spaces and equipped with a natural topology called the topology of the product ^[1] ^[2] or Tikhonov topology . The word “natural” is used here in the sense of category theory and means that this topology satisfies some universal property .

This topology was first investigated by the Soviet mathematician Andrei Tikhonov in 1926 .

Content

Definitions

Let be:

\{X_{\alpha }:\alpha \in A\}

{\ displaystyle \ {X _ {\ alpha}: \ alpha \ in A \}}

- family of topological spaces,

X=\prod \limits _{\alpha \in A}X_{\alpha }

{\ displaystyle X = \ prod \ limits _ {\ alpha \ in A} X _ {\ alpha}}

- their Cartesian product (as sets),

p_{\alpha }:X\to X_{\alpha }

{\ displaystyle p _ {\ alpha}: X \ to X _ {\ alpha}}

- the projection of the product on the corresponding factor.

Tikhonov topology on $X$ ${\ displaystyle X}$ Is the roughest topology (i.e., the topology with the least number of open sets ) for which all projections $p_{\alpha }$ ${\ displaystyle p _ {\ alpha}}$ continuous . The open sets of this topology are all possible unions of sets of the form $\prod _{i\in I}U_{i}$ ${\ displaystyle \ prod _ {i \ in I} U_ {i}}$ where each $U_{i}$ ${\ displaystyle U_ {i}}$ is an open subset $X_{i}$ ${\ displaystyle X_ {i}}$ and $U_{i}\neq X_{i}$ ${\ displaystyle U_ {i} \ neq X_ {i}}$ only for a finite number of indices. In particular, open product sets of finitely many spaces are simply the union of products of open subsets of the source spaces.

The Tikhonov topology can also be described as follows: as a prebase of the topology on $X$ ${\ displaystyle X}$ the family of sets is taken ${\mathfrak {P}}=\{p_{\alpha }^{-1}(U):\alpha \in A,\,U\in {\mathfrak {T}}_{\alpha }\}$ ${\ displaystyle {\ mathfrak {P}} = \ {p _ {\ alpha} ^ {- 1} (U): \ alpha \ in A, \, U \ in {\ mathfrak {T}} _ {\ alpha} \}}$ . Topology base - all kinds of finite intersections of sets from ${\mathfrak {P}}$ ${\ displaystyle {\ mathfrak {P}}}$ , and topology - all kinds of union sets from the base.

The Tikhonov topology is weaker than the so-called “boxed” topology, for which all kinds of products of open subsets of multiplied spaces form the base of the topology. Such a topology does not possess the universal property indicated above and the Tikhonov theorem is not true for it..

Examples

Normal topology on $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ( metric- induced topology) is the topology of the product on the Cartesian degree $\mathbb {R} .$ ${\ displaystyle \ mathbb {R}.}$

The Cantor set is homeomorphic to the product of a countable number of copies of a discrete space {0,1}, and the space of irrational numbers is a product of a countable number of spaces of natural numbers (with a discrete topology).

Properties

Topological space $X$ ${\ displaystyle X}$ together with projections on each component $X_{i}$ ${\ displaystyle X_ {i}}$ can be determined using the universal property : if $Y$ ${\ displaystyle Y}$ - arbitrary topological space and for each $i\in I$ ${\ displaystyle i \ in I}$ continuous display set $Y\to X_{i},$ ${\ displaystyle Y \ to X_ {i},}$ then there is a single mapping $Y\to X,$ ${\ displaystyle Y \ to X,}$ such that for each $i\in I$ ${\ displaystyle i \ in I}$ the following diagram is commutative:

This shows that the Tikhonov product is a product in the category of topological spaces . It follows from the universal property that the map $f:Y\to X$ ${\ displaystyle f: Y \ to X}$ continuous if and only if each mapping is continuous $f_{i}=p_{i}\circ f,$ ${\ displaystyle f_ {i} = p_ {i} \ circ f,}$ in many situations, continuity $f_{i}$ ${\ displaystyle f_ {i}}$ checking is easier.

Projections $p_{i}$ ${\ displaystyle p_ {i}}$ are not only continuous, but also (that is, each open set of a product, when projected onto a component, becomes an open set). The converse is generally not true (a counterexample is a subset of $\mathbb {R} ^{2},$ ${\ displaystyle \ mathbb {R} ^ {2},}$ complementing the open circle). Also, projections are not necessarily closed mappings (a counterexample is images of projections of a closed set $\{(x,y)\in \mathbb {R} ^{2}\mid xy=1\},$ ${\ displaystyle \ {(x, y) \ in \ mathbb {R} ^ {2} \ mid xy = 1 \},}$ on the coordinate axes are not closed subsets of the line).

The product topology is sometimes called the pointwise convergence topology. The reason for this is as follows: a sequence of elements from a product converges if and only if its image, when projected onto each component, converges. For example, the topology of a product on $\mathbb {R} ^{I},$ ${\ displaystyle \ mathbb {R} ^ {I},}$ space of real-valued functions on $I,$ ${\ displaystyle I,}$ Is a topology in which a sequence of functions converges when it converges pointwise.

Relationship with other topological concepts

Axioms of separability :

Composition $T_{0}$ ${\ displaystyle T_ {0}}$ -spaces has the property $T_{0}$ ${\ displaystyle T_ {0}}$ .
Composition $T_{1}$ ${\ displaystyle T_ {1}}$ -spaces has the property $T_{1}$ ${\ displaystyle T_ {1}}$ .
The product of Hausdorff spaces is Hausdorff.
The product of regular spaces regularly.
The product of completely regular spaces is quite regular.
The product of normal spaces is not always normal .

Compactness :

The product of compact spaces is compact .
The product of locally compact spaces is not always locally compact. However, the product of a family of locally compact spaces in which all components except a finite number are compact is locally compact.

Connectivity :

The product of connected (respectively, linearly connected) spaces is connected (respectively, linearly connected).
The product of completely disconnected spaces is completely disconnected.

Compactness of Tikhonov's works

Tikhonov's theorem : if all sets $X_{\alpha }$ ${\ displaystyle X _ {\ alpha}}$ are compact , then their Tikhonov product is compact.

To prove the statement, according to Alexander’s prebase theorem , it suffices to prove that any covering with elements of the prebase ${\mathfrak {P}}$ ${\ displaystyle {\ mathfrak {P}}}$ admits finite subcovering. For everyone $\alpha$ ${\ displaystyle \ alpha}$ let be $V_{\alpha }$ ${\ displaystyle V _ {\ alpha}}$ - union of all sets $U\in X_{\alpha }$ ${\ displaystyle U \ in X _ {\ alpha}}$ for which many $\pi _{\alpha }^{-1}(U)$ ${\ displaystyle \ pi _ {\ alpha} ^ {- 1} (U)}$ contained in the coating. Then the uncovered part of the space X is expressed by the formula:

\prod \limits _{\alpha \in A}X_{\alpha }\setminus V_{\alpha }

{\ displaystyle \ prod \ limits _ {\ alpha \ in A} X _ {\ alpha} \ setminus V _ {\ alpha}}

.

Since this set is empty, at least one factor must be empty. This means that the coating under consideration for some $\alpha$ ${\ displaystyle \ alpha}$ contains $\pi _{\alpha }$ ${\ displaystyle \ pi _ {\ alpha}}$ - prototype of the space cover $X_{\alpha }$ ${\ displaystyle X _ {\ alpha}}$ . Due to the compactness of space $X_{\alpha }$ ${\ displaystyle X _ {\ alpha}}$ , from its covering we can distinguish a finite subcover, and then its inverse image with respect to the mapping $\pi _{\alpha }$ ${\ displaystyle \ pi _ {\ alpha}}$ will be the final subcover of space $X$ ${\ displaystyle X}$ .

Notes

↑ Yu. G. Borisovich, N. M. Bliznyakov, T. N. Fomenko. Introduction to the topology. 2nd ed., Ext. - M .: Science. Fizmatlit., 1995. ISBN 5-02-014118-6 . S. 107.
↑ O. Ya. Viro, O.A. Ivanov, N. Yu. Netsvetaev, V.M. Kharlamov. Elementary topology. - M.: MCCNMO, 2012 .-- ISBN 978-5-94057-894-9 . S. 158.

Literature

Engelking R. General Topology. - M .: Mir , 1986 .-- 752 p.

[1] Yu. G. Borisovich, N. M. Bliznyakov, T. N. Fomenko. Introduction to the topology. 2nd ed., Ext. - M .: Science. Fizmatlit., 1995. ISBN 5-02-014118-6 . S. 107.

[2] O. Ya. Viro, O.A. Ivanov, N. Yu. Netsvetaev, V.M. Kharlamov. Elementary topology. - M.: MCCNMO, 2012 .-- ISBN 978-5-94057-894-9 . S. 158.