The answer is yes, a quotient map is closed. This means that the image of a closed set under a quotient map is also closed.
To prove that a quotient map is closed, we need to show that the image of a closed set under the quotient map is also closed. This can be done by using the definition of a quotient map and showing that the preimage of the complement of the image is open.
A quotient map is a type of function in mathematics that maps one set onto another, while preserving the structure of the original set. It is often used in topology to define the quotient space of a topological space by identifying certain points or sets together.
Yes, consider the function f:ℝ → ℝ/ℤ, where ℝ is the set of real numbers and ℝ/ℤ is the set of real numbers modulo the integers. This function maps each real number to its equivalence class in ℝ/ℤ. This is an example of a quotient map as it preserves the structure of the real numbers while mapping them onto the quotient space ℝ/ℤ.
Quotient maps are often used in topology to define the quotient space of a topological space. This is done by identifying certain points or sets together and then using the quotient map to map the original space onto the quotient space. Quotient maps are also important in understanding topological properties such as continuity and compactness.