Product Between R N \mathbb{R}^n R N And C \mathbb{C} C Is A Open Map

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Introduction

In the realm of general topology, a fundamental concept is the notion of an open map. An open map is a function between topological spaces that maps open sets to open sets. In this article, we will explore the product map between $\mathbb{R}^n$ and $\mathbb{C}$, denoted as $\cdot: \mathbb{R}^n \times \mathbb{C} \to \mathbb{C}^n$, and determine whether it is an open map.

The Product Map

The product map $\cdot: \mathbb{R}^n \times \mathbb{C} \to \mathbb{C}^n$ is defined as $\cdot((x_1,\dots,x_n),z)=(x_1 z,\dots,x_nz)$. This map takes a pair of an $n$-tuple of real numbers and a complex number, and outputs an $n$-tuple of complex numbers.

Understanding Open Maps

To determine whether the product map is an open map, we need to understand the concept of an open map. An open map is a function $f: X \to Y$ between topological spaces that satisfies the following property:

• For every open set $U \subseteq X$, the image $f(U)$ is an open set in $Y$.

In other words, an open map preserves the openness of sets.

The Product Map is an Open Map

To show that the product map is an open map, we need to prove that for every open set $U \subseteq \mathbb{R}^n \times \mathbb{C}$, the image $\cdot(U)$ is an open set in $\mathbb{C}^n$.

Let $U$ be an open set in $\mathbb{R}^n \times \mathbb{C}$. We need to show that $\cdot(U)$ is an open set in $\mathbb{C}^n$. To do this, we can use the definition of the product topology on $\mathbb{R}^n \times \mathbb{C}$.

The Product Topology

The product topology on $\mathbb{R}^n \times \mathbb{C}$ is defined as the coarsest topology such that the projection maps $\pi_i: \mathbb{R}^n \times \mathbb{C} \to \mathbb{R}$ and $\pi_j: \mathbb{R}^n \times \mathbb{C} \to \mathbb{C}$ are continuous for all $i$ and $j$.

Open Sets in the Product Topology

An open set in the product topology on $\mathbb{R}^n \times \mathbb{C}$ is a set that can be written as a union of sets of the form $\prod_{i=1}^n U_i \times V$, where $U_i$ is an open set in $\mathbb{R}$ and $V$ is an open set in $\mathbb{C}$.

The Image of an Open Set

Let $U$ be an open set in $\mathbb{R}^n \times \mathbb{C}$, written a union of sets of the form $\prod_{i=1}^n U_i \times V$. We need to show that $\cdot(U)$ is an open set in $\mathbb{C}^n$.

The Image of a Basis Element

Let $\prod_{i=1}^n U_i \times V$ be a basis element of $U$. We need to show that $\cdot(\prod_{i=1}^n U_i \times V)$ is an open set in $\mathbb{C}^n$.

The Image of a Basis Element is Open

The image of a basis element $\prod_{i=1}^n U_i \times V$ under the product map is given by:

$$\cdot(\prod_{i=1}^n U_i \times V) = \prod_{i=1}^n (U_i \cdot V)$$

where $U_i \cdot V$ is the set of complex numbers of the form $u_i \cdot z$, where $u_i \in U_i$ and $z \in V$.

The Set $U_i \cdot V$ is Open

The set $U_i \cdot V$ is an open set in $\mathbb{C}$ because it is the image of the open set $U_i \times V$ under the continuous map $(u_i, z) \mapsto u_i \cdot z$.

The Product of Open Sets is Open

The product of open sets is open. Therefore, the product $\prod_{i=1}^n (U_i \cdot V)$ is an open set in $\mathbb{C}^n$.

Conclusion

We have shown that the product map $\cdot: \mathbb{R}^n \times \mathbb{C} \to \mathbb{C}^n$ is an open map. This means that for every open set $U \subseteq \mathbb{R}^n \times \mathbb{C}$, the image $\cdot(U)$ is an open set in $\mathbb{C}^n$.

References

• [1] Munkres, J. R. (2000). Topology. Prentice Hall.
• [2] Kelley, J. L. (1955). General Topology. Springer-Verlag.

Further Reading

• [1] Bourbaki, N. (1966). General Topology. Springer-Verlag.
• [2] Willard, S. (1970). General Topology. Addison-Wesley.

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Introduction

In our previous article, we explored the product map between $\mathbb{R}^n$ and $\mathbb{C}$, denoted as $\cdot: \mathbb{R}^n \times \mathbb{C} \to \mathbb{C}^n$, and determined that it is an open map. In this article, we will answer some frequently asked questions about the product map and its properties.

Q: What is the product map?

A: The product map $\cdot: \mathbb{R}^n \times \mathbb{C} \to \mathbb{C}^n$ is a function that takes a pair of an $n$-tuple of real numbers and a complex number, and outputs an $n$-tuple of complex numbers.

Q: Why is the product map an open map?

A: The product map is an open map because it preserves the openness of sets. Specifically, for every open set $U \subseteq \mathbb{R}^n \times \mathbb{C}$, the image $\cdot(U)$ is an open set in $\mathbb{C}^n$.

Q: What is the significance of the product map being an open map?

A: The product map being an open map has significant implications for the study of topological spaces. It means that the product of two topological spaces is a topological space in its own right, and that the product map preserves the topological structure of the spaces.

Q: Can you provide an example of the product map?

A: Yes, consider the product map $\cdot: \mathbb{R}^2 \times \mathbb{C} \to \mathbb{C}^2$ defined by $\cdot((x_1, x_2), z) = (x_1 z, x_2 z)$. This map takes a pair of a 2-tuple of real numbers and a complex number, and outputs a 2-tuple of complex numbers.

Q: How does the product map relate to other topological concepts?

A: The product map is closely related to other topological concepts, such as the product topology and the Tychonoff theorem. The product topology is a topology on the product of two topological spaces, and the Tychonoff theorem states that the product of compact spaces is compact.

Q: Can you provide a proof of the product map being an open map?

A: Yes, the proof of the product map being an open map is as follows:

Let $U$ be an open set in $\mathbb{R}^n \times \mathbb{C}$. We need to show that $\cdot(U)$ is an open set in $\mathbb{C}^n$. To do this, we can use the definition of the product topology on $\mathbb{R}^n \times \mathbb{C}$.

Let $\prod_{i=1}^n U_i \times V$ be a basis element of $U$. We need to show that $\cdot(\prod_{i=1}^n U_i \times V is an open set in $\mathbb{C}^n$.

The image of a basis element $\prod_{i=1}^n U_i \times V$ under the product map is given by:

$$\cdot(\prod_{i=1}^n U_i \times V) = \prod_{i=1}^n (U_i \cdot V)$$

where $U_i \cdot V$ is the set of complex numbers of the form $u_i \cdot z$, where $u_i \in U_i$ and $z \in V$.

The set $U_i \cdot V$ is an open set in $\mathbb{C}$ because it is the image of the open set $U_i \times V$ under the continuous map $(u_i, z) \mapsto u_i \cdot z$.

The product of open sets is open. Therefore, the product $\prod_{i=1}^n (U_i \cdot V)$ is an open set in $\mathbb{C}^n$.

Conclusion

We have answered some frequently asked questions about the product map and its properties. The product map is an open map because it preserves the openness of sets, and it has significant implications for the study of topological spaces.

References

• [1] Munkres, J. R. (2000). Topology. Prentice Hall.
• [2] Kelley, J. L. (1955). General Topology. Springer-Verlag.

Further Reading

• [1] Bourbaki, N. (1966). General Topology. Springer-Verlag.
• [2] Willard, S. (1970). General Topology. Addison-Wesley.