91Ó°ÊÓ

A topological space is one of the fundamental constructs in topology. It consists of a set \(X\) along with a collection of open sets \(\jmath\) that satisfy certain properties. These properties include:

• The entire set \(X\) and the empty set \(\emptyset\) are included in the collection \(\jmath\).
• The intersection of any finite number of these open sets is also an open set.
• The union of any collection of these open sets is also in \(\jmath\).

Topological spaces provide a way to examine the concept of continuity, limits, and convergence without the need for a distance metric. Although they are abstract, visualizing simple examples like the real numbers with the standard topology can help. Here, open intervals, such as \((a, b)\), serve as the open sets. For a space to be metrizable, there must exist a metric—a function that defines a distance—which induces its topology.

In topological spaces, a neighborhood of a point \(a\in X\) is a set that includes an open set containing \(a\). The idea of a neighborhood extends beyond a specific point and helps in understanding concepts such as continuity. Here’s what makes a neighborhood special:

• Every open set containing \(a\) is a neighborhood of \(a\).
• A neighborhood \(N\) of a point \(a\) does not have to be open itself.

For instance, an open interval \((a - \varepsilon, a + \varepsilon)\) in the real numbers is a neighborhood of \(a\). In our example, we can choose an open set \(U\) that is part of \(N\) and contains the point \(a\), ensuring the property holds true. Neighborhoods are essential for the definitions of limit points and continuity.

The concept of an open ball is a central idea in metric spaces, which are a special kind of topological space. An open ball around a point \(a\in X\) with radius \(\varepsilon > 0\) is defined as \(B(a, \varepsilon) = \{x \in X \mid d(x, a) < \varepsilon\}\). In a metric space, this open ball helps define the open sets that form the topology.
Open balls have specific qualities:

• They are always open sets by definition.
• A point \(a\) is always in the open ball centered at itself.

In the context of the given problem, since our space is metrizable, we utilize these balls to find closed neighborhoods by taking closures of open balls. An open ball like \(B(a, \varepsilon)\) offers a clear visual representation and foundation for observing continuity and limits.

Closure is a concept used to understand the "bounded" nature of a set in topological terms. For a set \(S\) within a topological space, the closure, denoted \(\overline{S}\), is the smallest closed set containing \(S\). It includes:

• All points in \(S\),
• as well as all limit points of \(S\).

The closure operation is crucial in determining the "boundary" points of a set and plays a significant role in defining continuity.
In metric spaces, closure can be directly connected to sequences: a point is in the closure if there is a sequence in \(S\) converging to that point. In our problem, by considering the closure of an open ball, such as \(\overline{B(a, \varepsilon/2)}\), we ensure \(a\) is within a closed set that serves the necessary properties for our topological proofs.