A cover of a set X is a collection Σ of subsets of X whose union is the whole X. A cover ∑′ is called subcover of a cover Σ provided each of sets from the collection ∑′ belongs to Σ. If each cover of a topological space X by open sets contains a finite subcover then the space X is called compact.
The compactness of a space X is equivalent to each of the following conditions:
1)each net in X contains a convergent subnet;
2)each centered collection of closed subsets of X (i.e., such collection that each its nonempty finite subcollection has a nonempty intersection) also has a nonempty intersection.
A set X is said to be relatively compact, if its closure
By virtue of the Tychonoff theorem the topological product X1 × X2 of compact spaces X1 and X2 is compact.
A subset of the Euclidean n-dimensional space
A cover Σ of a topological space X is called locally finite if every point x ∈ X possesses a neighborhood U which intersects only a finite number of sets from Σ. A topological space X is said to be paracompact if it is Hausdorff and each its open cover Δ has an open locally finite refinement Σ (i.e., each of the sets from Σ is contained in a set from Δ).
For each locally finite open cover Ξ = {Ui}j∈J of a paracompact space X there exists a subordinated partition of unity, i.e., a family {pj}j∈J of continuous on X nonnegative functions such that:
1)for each j ∈ J we have: supp pj = {x|x ∈ X, pj(x) ≠ 0} ⊂ Uj;
2)for each point
Notice that due to the local finiteness of the cover Ξ, only a finite number of terms in the last sum differ from zero.
Let (X, ϱ) be a metric space, x ∈ X, and r > 0. The set
is called an open ball of the radius r with the center at x, whereas the set
is a closed ball of the radius r with the center at x. The collection of all open balls is the base of a certain topology on X which is called metric topology. It is clear that a set V in a metric space X is open if and only if every point x of V belongs to V with a certain open ball centered at x.
Two metrics on a set X are called equivalent if they generate on X the same metric topology. Each space with metric topology is normal and hence regular and Hausdorff. Due to the Stone theorem every metric space is paracompact.
The distance from a point x to a set A ⊂ X is defined as
If A ⊂ X and ε > 0 then the set
is called an ε-neighborhood of the set A.
Let A be a compact metric space and Σ an open cover of X. Then, according to the Lebesgue covering lemma (see, e.g., [241]), there exists a positive real number r with the property that for each x ∈ X there exists a set U ∈ Σ such that Br(x) ⊂ U. From this assertion it follows that if (X, ϱ) is a metric space, A is a compact subset of X, B is a closed subset of X and A ⋂ B =
and hence there exists such ε > 0 that
This yields, in particular, that for every open neighborhood U of a compact set A there exists a sufficiently small ε > 0 such that the ε-neighborhood of A is contained in U.
Let T be a compact space, (X, ϱ) a metric space. On the set C(T, X) of all continuous functions from T to X the matric
The topology τc generated on C(T, X) by this metric is called the topology of uniform convergence.
Let (T, ϱT) be a compact metric space. A family of functions H ⊂ C(T, X) is called equicontinuous if for each ε > 0 there exists δ > 0 such that for every t, t′ ∈ T the condition ϱT(t, t′) < δ implies ϱ(f(t), f(t′)) < ε for all f ∈ H. According to the Arzela–Ascoli theorem (see, e.g., [241]) if a subset H ⊂ C(T, X) is equicontinuous and the sets H(t) = {f(t) |f ∈ H} are relatively compact in X for all t ∈ T then H is relatively compact in the space (C(T, X), τc).
If X is a linear space and A, B ⊂ X then
If α ∈
The set of all finite linear combinations
where
Let X be a linear space on which a topology τ be defined. The pair (X, τ) is said to be a linear