Preimage Problem: Understand Closure & Metrics

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In summary, a function f:X\to Y with sets X and Y and a subset B\subset Y has a preimage f^*(B) = \{x\in X : f(x) \in B\}. This preimage can be characterized by the continuity of f. The preimage of any open (resp. closed) set in (Y,d_Y) is open (resp. closed) in (X,d_X). For example, the function f_1(x,y)=x-y is continuous and the set A_1=f_1^*\left( (-\infty ,1] \right) is closed. Similarly, A_2 and A_3 are also closed, but it is not clear
  • #1
Ted123
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[PLAIN]http://img855.imageshack.us/img855/5949/metric.jpg

If [itex]X,Y[/itex] are sets and [itex]f:X\to Y[/itex] is a function with [itex]B\subset Y[/itex], the preimage is defined [itex]f^*(B) = \{x\in X : f(x) \in B\}[/itex].

If [itex]d_X, d_Y[/itex] are metrics on [itex]X,Y[/itex], continuity of [itex]f[/itex] can be characterised as follows:

The preimage of any open (resp. closed) set in [itex](Y,d_Y)[/itex] is open (resp. closed) in [itex](X,d_X)[/itex].

Hence, for example if we define [itex]f_1 (x,y) = x-y[/itex] then [itex]f_1[/itex] is continuous and [itex]A_1 = f_1^*\left( (-\infty ,1] \right)[/itex]. Since [itex](-\infty , 1][/itex] is closed, [itex]A_1[/itex] is closed.

Similarly for [itex]A_2[/itex] and [itex]A_3[/itex], but not sure about [itex]A_4[/itex]. Can I write it in a way that makes it more obvious/easier to work with the preimage?
 
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  • #2


Left f(x,y)= x- y, a continuous function. The set of all (x, y) such that [itex]x- y\le 1[/itex] is the preimage, by that function, of the set [itex]\{z| z\le 1\}[/itex]. Since that is a closed interval, it follows that A1 is a closed sert.
 
  • #3


I know that ([itex]A_1[/itex] is the one I proved already!) - it's [itexA_4[/tex] that I can't see how to do...
 

1. What is the preimage problem?

The preimage problem is a mathematical concept that involves understanding the closure and metrics of a set. It involves finding the input or "preimage" of a function that produces a specific output or "image".

2. Why is the preimage problem important?

The preimage problem is important because it helps us understand the properties and behavior of functions and sets. By understanding the closure and metrics of a set, we can analyze the convergence, continuity, and other important characteristics of a function.

3. What is closure in relation to the preimage problem?

Closure refers to the property of a set that contains all of its limit points. In the preimage problem, closure is important because it helps us determine if a function has a well-defined preimage for a given image.

4. How do metrics relate to the preimage problem?

Metrics, or distance functions, are used to measure the distance between points in a set. In the preimage problem, metrics help us determine if a function satisfies certain properties, such as continuity or convergence.

5. What are some common strategies for solving the preimage problem?

Some common strategies for solving the preimage problem include using inverse functions, using algebraic manipulations to isolate the input variable, and using geometric interpretations to visualize the problem. Additionally, understanding the properties of closure and metrics can also aid in solving the preimage problem.

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