Question

Let \(E\) be a topological linear space. Prove that a) If Uand Vare open sets, then so is \(U+V=\\{z: z=x+y, x \in U\), \(y \in V\\}\) b) If \(U\) is open, then so is \(\alpha U=\\{z: z=\mathrm{ax}, \mathrm{x} \in U\\}\) provided that \(\mathrm{a} \neq 0\) c) If \(F \subset E\) is closed, then so is \(\alpha F\) for arbitrary a.

Short answer

a) \(U+V\) is open; b) \(\alpha U\) is open for \(\alpha \neq 0\); c) \(\alpha F\) is closed.

Step-by-step solution

Define the Sum of Open Sets

To show that the sum of two open sets \(U\) and \(V\) in a topological linear space \(E\) is open, we need to consider arbitrary points \(x \in U\) and \(y \in V\). We want to show that any point \(z = x + y\) in \(U + V\) has a neighborhood entirely contained within \(U + V\).

Use the Openness Property of Topological Spaces

Since \(U\) and \(V\) are open, there exist open neighborhoods \(U_x \subset U\) and \(V_y \subset V\) around \(x\) and \(y\) respectively. The sum \(U_x + V_y\) is then an open neighborhood of \(z = x + y\) and is a subset of \(U + V\), showing \(U + V\) is open.

Define Scaling of an Open Set

To demonstrate that \(\alpha U\) is open when \(U\) is open and \(\alpha eq 0\), consider any element \(z = \alpha x\) in \(\alpha U\), where \(x \in U\). We need to find a neighborhood of \(z\) entirely contained within \(\alpha U\).

Use Algebraic Properties of Topology

Since \(U\) is open and contains \(x\), there exists an open neighborhood \(U_x\) around \(x\). Then the set \(\alpha U_x\) forms an open neighborhood of \(z = \alpha x\) and is a subset of \(\alpha U\), proving \(\alpha U\) is open.

Define Scaling of a Closed Set

For a closed set \(F\) in \(E\), we need to show that \(\alpha F\) is closed when \(\alpha\) is any scalar. Recall that the image of a closed set under a continuous function is closed.

Use Continuity and Linear Properties

The map \(x \mapsto \alpha x\) is continuous for any scalar \(\alpha\). Thus, the image of a closed set \(F\) under this mapping, \(\alpha F\), is also closed, proving the statement.

Key concepts

Understanding Open Sets in Topological Spaces

An open set in a topological linear space is a foundational concept in topology. Simply put, a set is open if for every point within the set, there exists a neighborhood completely contained in that set.
This means that no matter where you look in the set, you can find a small surrounding area (a neighborhood) that doesn't stray outside the set.

When exploring open sets in mathematical contexts, particularly in a topological linear space, we encounter interesting properties.
For example, the sum of two open sets is still open. Imagine you have two open sets, say a circle and a square, within a space.

• The addition of these two sets involves combining all possible sums of their elements.
  The resultant set still shares this openness because any point in this new sum can find itself surrounded by a neighborhood within the sum of the sets, ensuring the entire set remains open.

Exploring Closed Sets

Closed sets are essentially the complement of open sets. In a given topological space, a set is closed if its complement is open.
A closed set includes its boundary points and is robust under limit operations - meaning that any limit of a sequence of points in a closed set must also be in that set.

Consider what happens when you scale a closed set by multiplying all its points by a scalar number, whether positive or negative.

• Interestingly, this process of scaling doesn't affect the closed nature of the set.
  This is because the function of scaling is continuous in topology, and the image of a closed set under a continuous function is also closed.
  So, even if you multiply the set by any number, it retains its original closed properties.

The Concept of a Neighborhood

In the language of topology, a neighborhood refers to a "bubble" of space surrounding a particular point.
It's not just an isolated point, but rather a collection of points that cluster closely around it.

When we say a point is in the neighborhood, it implies that within a certain radius around this point, all other points belong to the same set.

• Neighborhoods are crucial for understanding open sets because a set is open if for every point, you can find such a neighborhood.
  These localized collections help us decipher whether a point can access its nearby neighbors without stepping outside the confines of the set.

Talking about these neighborhoods lets us dive deeper into analyzing the properties of sets in topological spaces, offering insights into complex mathematical relationships.

Scaling of Sets in Topological Spaces

Scaling of sets is an intriguing operation in a topological space, particularly when dealing with both open and closed sets.
It involves multiplying all elements of a set by a constant scalar, which geometrically stretches or shrinks the set.

When we focus on open sets:

• If you scale an open set by a nonzero constant, it remains open.
  That's because you can continuously stretch or shrink the space without altering the essential property that makes it open.
  Every point within the scaled set will have its neighborhood that still fits entirely in the scaled set.

On the other hand, when scaling a closed set:

• The result is a closed set under any scalar multiplication.
  This can be attributed to the continuous nature of scaling transformations, which keep the closed property intact.
  Whether you magnify or condense the set, its characteristic as a closed set persists.

These properties highlight the dynamic interactions within topological spaces, making scaling a powerful tool to understand and visualize transformations within these mathematical structures.