Csaba Vincze (2013)

University of Debrecen

Let r be a positive real number. The open ball around the point p with radius r is defined as the set of points all of whosedistance from p is less than r. A subset U in the space is open if for any point p in U is contained together with an open ball around p. In other words p is an interior point. A subset is closed if its complement is open. It can be easily seen that

(T1) both the empty set and the entire space are open (and, at the same time, they are closed).

(T2) the union of the elements of an arbitrary family of open subsets is open.

(T3) the intersection of finitely many open subsets is open.

In general the family of subsets satisfying conditions (T1)-(T3) is called topology. The members of the topology are the open subsets. The topology has a countable basis if there exists a countable collection

${U}_{1},{U}_{2},\mathrm{\dots},{U}_{n},{U}_{n+1},\mathrm{\dots}$ |
(1.14) |

of open subsets such that for any open subset can be written as the union of the elements of some subcollection. It is just the second axiom of countability and the space equipped with a topology having a countable basis is called second-countable space.

**Example **The Euclidean space of dimension n is a second countable space because the collection of open balls having centers with rational coordinates and
positive rational numbers as radiuses forms a basis for the usual topology.

An open cover of a subset A is a family of open subsets containing A in the union of its elements. The subset A is compact if every open cover contains a finite subcover. It is known (see Heine-Borel theorem) that the compactness is equivalent to the boundedness and closedness in the real coordinate spaces. A subset is bounded ifit is contained in an open ball around the origin with a finite radius.

**Definition **The closure of a subset A in a topological space is the intersection of closed subsets containing A. The interior of A is the union of open subsets contained in A.

**Theorem 1.2.1 **
*(Lindelöf, Ernst Leonard) Every open cover in a second-countable space contains a countable subcover. *

• Consider an arbitrary open cover of the subset A in a second-countable topological space:

$A\subset \bigcup _{\gamma \in \mathrm{\Gamma}}{V}_{\gamma}=\bigcup _{\gamma \in \mathrm{\Gamma}}\left(\bigcup _{i\in {I}_{\gamma}}{U}_{i}\right),$ |
(1.15) |

where U(1), ..., U(m), ... is a basis of the topology and

${V}_{\gamma}=\bigcup _{i\in {I}_{\gamma}}{U}_{i},\text{}where\text{}{I}_{\gamma}\subset N.$ |
(1.16) |

If I is the union of I(γ) as γ runs through the set Γ then it is a countable set of indices. Equation 1.15 shows that A is a subset in the union of U(i)'s as i runs through the set I. Since for any i there exists γ(i) such that U(i) is a subset in V(γ(i)) we have that

$A\subset \bigcup _{i\in I}{V}_{{\gamma}_{i}}$ |

and the subcollection

${V}_{{\gamma}_{1}},{V}_{{\gamma}_{2}},\mathrm{\dots},$ |

is a countable open subcover for the subset A as was to be proved. ▮