Projection Map $X \times Y$: Closure Property

In summary, the closure property of a projection map refers to the fact that the image of the map will contain all possible combinations of second coordinates from the original product set. This property is used in various mathematical concepts and proofs, and is important in guaranteeing a comprehensive representation of the original set. It can be proven using mathematical techniques and is closely related to the concept of surjectivity.
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Euge
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Here is this week's POTW:

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Let $X$ and $Y$ be topological spaces. If $Y$ is compact, show that the projection map $p_X : X \times Y \to X$ is closed.
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No one solved this problem. You can read my solution below.

Let $C$ be a closed set in $X \times Y$. If $x\in X\setminus p_X(C)$, then for every $y\in Y$ the ordered pair $(x,y)\notin C$; it follows that there are open neighborhoods $U_y$ of $x$ and $V_y$ of $y$ such that $(U_y\times V_y)\cap C = \emptyset$. The collection $\{V_y:y\in Y\}$ is an open cover of $Y$; by compactness of $Y$, there are $y_1,\ldots, y_n\in Y$ such that $Y = V_{y_1}\cap \cdots \cap V_{y_n}$. Let $U = U_{y_1}\cap \cdots \cap U_{y_n}$. Then $U$ is an open neighborhood of $x$ such that $(U\times Y) \cap C = \emptyset$, i.e., $p_X^{-1}(U)\cap C = \emptyset$. Thus $U \cap p_X(C) = \emptyset$. Since $x$ was arbitrary, $p_X(C)$ is closed.
 

Related to Projection Map $X \times Y$: Closure Property

1. What is the closure property of a projection map?

The closure property of a projection map refers to the fact that when a projection map is applied to a set of points in a Cartesian product, the resulting set of points will also be in the Cartesian product. In other words, the projection map preserves the structure of the Cartesian product.

2. How does the closure property of a projection map relate to mathematical operations?

The closure property of a projection map is similar to the closure property of mathematical operations such as addition and multiplication. Just like how adding or multiplying two numbers results in another number, applying a projection map to a set of points in a Cartesian product results in another set of points in the same Cartesian product.

3. Why is the closure property important in mathematics?

The closure property is important in mathematics because it allows for the creation of new mathematical structures and operations. For example, the closure property of a projection map allows for the creation of new sets and operations on those sets, which can be useful in solving mathematical problems and proving theorems.

4. Can a projection map have the closure property for more than two sets?

Yes, a projection map can have the closure property for any number of sets. The closure property simply means that the resulting set after applying the projection map will still be in the Cartesian product of the original sets, regardless of how many sets are involved.

5. What are some real-world applications of the closure property of a projection map?

The closure property of a projection map has many real-world applications, particularly in computer science and engineering. It is often used in data compression and coding, as well as in the design of computer algorithms and data structures. It can also be applied in physics and engineering, where Cartesian products are used to model physical systems.

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