Basic Concepts

On a set \(X\) is defined a topology if there is a class of its subsets that is (the class) stable under any union and any finite intersection. The sets in the given class are called open sets. As usual, two immediate topologies come in mind: the trivial topology consisting of only two open sets, and the discrete topology with all subsets of \(X\) being open sets.

The most fundamental concept in topology is undoubtedly the concept of neighborhood which provides the precise setting for what we ordinarily call near to.... The neighborhood of a point in a topological space is any open set containing that point. Other concepts include:

• Interior. The interior (part) of \(A\), \(Int(A)\) or \(Int_X(A)\) is the largest among the open sets contained in \(A\): \( Int(A) = \cup_{\mathcal{O}\, \text{open},\mathcal{O}\subset A} \mathcal{O} \)


• Closure . The closure of \(A\), denoted by \(Cl(A)\), is the smallest among the closed sets containing \(A\), i.e. the intersection of all closed sets containing \(A\) or the closed set generated by \(A\);


• For all \(A\): \( Int(A) \subset A \subset Cl(A) ;\)
• Boundary. \( Fr A = \partial A = Cl(A) \setminus Int(A); \)
• The point in \( Int(A) \), \( Cl(A) \), \(Fr(A)\), and \( X \setminus Cl(A) = Int(X\setminus A) \) are called interior , adherent, boundary and exterior point respectively;
• A point is interior to \(A\) if there is an open set containing that point and entirely contained in \(A\): \(x \to Int(A)\) if there exists \(\mathcal{O}\) open set such that \(\mathcal{O}\) is a part of \(A\) and \(x\to \mathcal{O}\). (Notion of parts and containment are categorical.)
• A point is adherent to \(A\) if any arbitrary of its neighborhood intersects \(A\): \(x\to Cl(A)\) if for any open set \(\mathcal{O}\) containing \(x\), there exists \( x\neq y \to \mathcal{O}\cap A \);
• A point is boundary point to \(A\) if any arbitrary of its neighborhoods intersects both \(A\) and \( X\setminus A \); and finally
• A point is exterior point to \(A\) if it has a neighborhood that does not intersect \(A\).


• Everywhere dense. A subset \(A\) of \(X\) is everywhere dense in the top space \(X\) if \(A\) intersects any nonempty open set in \(X\), i.e. if its closure is the whole space \(X\). A nowhere dense subset is a subset whose complement is everywhere dense. Subsets of \(X\) can be classified according to whether they are countable union of nowhere dense subsets. Any subset that is so is of first category; any other is of the second category. In a completemetric space, the so-called Baire category theorem (BCT) states that if \(A\subset X\) is of first category, then \(X\setminus A\) is everywhere dense, that is of second category.

Bases and Prebases

We are now interested in some collection of subsets that allow us to generate the whole class topology. These are prebases and bases. For a top space \(X\), a basis \(\mathcal{B}\) is a collection of open sets such that any open set in \(X\) is presented as union of elements from \(\mathcal{B}\) A base is fully characterized by the following: for any two open sets \(U\) and \(V\) from \(\mathcal{B}\), and for any \(x \in U\cap V \), there always exists an open set \(W \in \mathcal{B}\) such that \( x\in W \subset U\cap V\): any two open sets that intersect at a point not only contain that point but even contain a whole open set containing that point. In topology, containing open sets for a subspace translates into its extensiveness. On the other hand, as we will see, being contained into a compact translates into some kind of narrowness of the subspace. As Villani said, open sets are those sets that can be viewed as large; compact sets are those ones that can be viewed as small.

We will se that balls in metric space are prototypical examples of top space bases.

Base vs Basis? I shall use base if there is no notion of independence; while basis shall be used when the family is both generating and independent. In our view, base is broader than basis.

Covers

If \(\Gamma\) is a collection of subsets of a topological space, \(\Gamma\) covers \(A \subset X \) if \( A \subset \cup_{i\in I} G_i \) where \(G_i\), \( i\in I\) are elements from the cover. We shall speak of:

• Refinement. \(\Gamma\) is a refinement of \(\Delta\) if any member of \(\Gamma\) is contained in an element of \(\Delta\);
• Local finiteness. The cover \(\Gamma\) is locally finite if if any point of the space has a neighborhood that intersects only a finite number of elements from \(\Gamma\);
• Open cover (closed cover). If the members are all open sets (all closed ssets).Every open cover has a refinement.

Metrics

1. Separation: \(\rho(x, y) = 0 \iff x = y\);
2. symmetry: \(\rho(y, x) = \rho(x, y)\);
3. Triangle inequality: \(\rho(x, y) \leq \rho(x, z) + \rho(z, y)\) \( \forall x\, \forall\,y \, \forall\, z \);

1. Open ball \( B_r(x) = \{ y\in X: d(x, y) < r\} \);
2. Closed ball \(B_{r]}(x) = \{ y \in X: d(x, y) \leq r \} \);
3. Sphere \(S_r(x) = \{ y \in X: d(x, y) = r \} \);

Examples of ball and sphere include the unit \(n-\)dimensional ball \(D^n\) and the unit \((n-1)-\)dimensional sphere \(S^{n-1}\) of \( \mathbb{R}^n \) centered around \((0, \dots, 0)\)

Distance between two subsets of a metric space: if \(A, B \subset (X, d)\) then \( dist(A, B) = \min\{ d(x, y): x\in A\,, y\in B \} \) and the diamter of \(A\) is defined as \(\operatorname{diam}(A) = \sup_{x, y\in A} d(x, y)\). A subset \(Y\) of a metric space \((X, d)\) is called bounded if \(\operatorname{diam}(Y)\) is finite.

Metric topology. If \( x_1 \in B_r(x_0) \) then \( B_{r_1}(x_1)\subset B_r(x_0) \) where \(r_1 = r - d(x_0, x_1)\). Therefore the intersection of two open balls contains another open ball of smaller radius. Moreover, open balls cover the entire space. Therefore, any metric space is a topological space: it has as a base the family of all open balls. This topology is called metric topology.

A top space whose topology is metric topology with respect to some metric is said to be metrizable.

Open balls centered at a given point of a metric space constitute a base at that point. The part of this base consisting of the balls of radii \(\frac{1}{n}\) for \( n =1, 2, 3, \dots\) is also a base.

Metric neighborhood: the metric neighborhood of a subset \(A\subset X\) is the set \( \mathcal{RA}(A) = \{ x\in X: \operatorname{dist}(A, x) < r \} = \bigcup_{x\in A} B_r(x) \).