Topology produced

The topology produced is a Topologie definite on a product of topological spaces. It is in a general way the weak Topologie associated with projections of space produces towards each one of its factors: in other words it is the least fine topology making projections continuous.

Case of the end product

In the case of the end product, topology produced in particular makes it possible to define a natural topology on $\ mathbb {R} ^n$ from that of $\ mathbb {R}$ .

If $X_1. X_n$ are topological spaces, $U$ is open of $X= \ prod_ {i=1} ^n X_i$ if and only if $\ forall X \ in U$ there exists $U_1. U_n$ open respective of $X_1. X_n$ such as $x \ in U_1 \ times. \ times U_n$ and $U_1 \ times. \times U_n\subset U$ . In other words open of the product is a meeting of products of open factors.

One can check that this definition makes projections continuous (one will see in the following part that this characterizes in fact topology produced), and that projected of open is open. On the other hand, projected of one closed is not closed. For example, the unit $\ mathcal {H} = \ {(X, there) \ in \ mathbb {R} ^2, xy=1 \}$ is closed of $\ mathbb {R} ^2$ (it is the reciprocal image of one closed by a function continues), but its projection on the x axis is not closed (it is indeed $\ mathbb {R} ^*$ ).

General case

Either $(X_i, \ tau_i) _ {I \ in I}$ an unspecified family of topological spaces, the product of the $X_i$ is noted $X$ . Topology produced is the least fine topology returning projections $p_i: X \ rightarrow X_i$ continuous: its prébase is thus the whole of the $p_i^ {- 1} (U_i)$ , $U_i$ open of $X_i$ , $i \ in I$ , in other words it is:

$\ {U_i \ times \ prod_ {J \ in I, J \ I} X_j, U_i \ in \ tau_i, I \ in I \}$ .

A base of topology produced is then formed by the whole of the finished intersections of elements of the prébase; by noticing that if $i \ j$ then $(U_i \ times \ prod_ {K \ in I, K \ I} X_k) \ bigcap (U_j \ times \ prod_ {K \ in I, K \ J} X_k) =U_i \ times U_j \ times \ prod_ {K \ in I, K \ not \ in \ {I, J \}} X_k$ , one sees whereas a base of topology produced is:

$\ {\ prod_ {k=1} ^n U_ {i_k} \ times \ prod_ {J \ in I, J \ not \ in \ {i_1. .i_n \}} X_j, U_ {i_1} \ in \ tau_ {i_1}. U_ {i_n} \ in \ tau_ {i_n}, i_1. .i_n \ in I, N \ in \ mathbb {NR} \}$

One then deduces easily the case finished by noticing that spaces $X_1. X_n$ are the open ones, and that reciprocally very produced the open ones of $X_1. X_n$ is inevitably finished! On the other hand in the case of the infinite product, the base consists of products of a number finished of open with remaining spaces: an infinite product of open is not a priori open .

Important properties

Is $X$ the product of the $X_i, I \ in I$ , $Y$ a topological space. $f: Y \ rightarrow X$ is continuous if $\ forall I \ in I, p_i \ circ F$ is continuous (to prove it, to use the characterization of continuity via a prébase of open).

Moreover, one notices that in topology produced, to say that $(x^n) _ {N \ in \ mathbb {NR}}$ tightens towards $y= (y_i) _ {I \ in I}$ is equivalent to: $\ forall I \ in I$ , $(x^n_i) _ {N \ in \ mathbb {NR}}$ tends towards $y_i$ . When the $X_i$ all are equal to $X_0$ , the product of the $X_i$ is in fact the whole of the applications of $I$ in $X_0$ (from where the notation $X_0^i$ ). The continuation $(x^n)$ preceding is then a succession of functions $f_n$ which converges simply towards the function $g: I \ mapsto y_i$ . One can summarize this by saying that topology produced is the topology of simple convergence .

Lastly, one of the most important theorems concerning topology produced is Théorème of Tychonov which ensures that a product of compact is compact for this topology.

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