# Cardinality of the Preimage f^{-1}(y) of f:X->Y continuous?

• Bacle2
In summary, the question being discussed is about the cardinality of the preimage of a non-constant, continuous function between topological spaces X and Y, given that the fiber is discrete and uncountable. Examples are given of fibers being both countably-infinite and uncountable, and a potential solution is suggested using a metric space and an uncountable subset with isolated points. Later, it is mentioned that the condition of neither topology being discrete or indiscrete is necessary, but this does not affect the potential solution. The conversation ends with the poster planning to delete their original post and asking for advice on how to do so.
Bacle2
Cardinality of the Preimage f^{-1}(y) of f:X-->Y continuous?

Hi, All:

Let X,Y be topological spaces and f:X-->Y non-constant continuous function.

I'm curious as to whether it is possible for the fiber {f^{-1}(y)} of some y in Y

to be uncountable, given that the fiber is discrete (this condition is intended to

discount cases like f:ℝ-->ℝ, that are 0 in some interval, using, e.g., combinations

of e-1/x2) . There are examples of fibers being countably-

infinite, like in f:ℝ→ ℝ, f(x)=sinx, cosx, etc. , or the topologists sine curve on [0,1].

I suspect it may be possible, if X is metric to use f:X-->X with f(x)=d(x,S) , i.e.,

S is a subset of X , and d(x,S):=inf{d(x,s): s in S}. I thought S=Cantor set may work, but

the points of C are not isolated in ℝ (in [0,1], actually). Maybe if one can define an uncountable subset of

a metric space , all of whose points are isolated, we would be done.

Any IDeas?

Last edited:

Let X be any uncountable discrete space and map it any way you want onto a two-point space. This will give a continuous map whose fibre above at least one of the points in the range will necessarily be uncountable...

Bacle2 said:
Maybe if one can define an uncountable subset of

a metric space , all of whose points are isolated, we would be done.

Think of the discrete metric on an uncountable number of points.

It may be interesting to notice that if the metric space is seperable, then every set of isolated points is countable. Indeed, let S be a set of isolated points. Since the metric space is seperable, it is second countable. So S is second countable and thus seperable (not that a subspace of a seperable space is not always seperable, but I've just shown that it is in metric spaces). But every point in S is isolated, so every singleton in S is open. It is clear that S must be countable.

Thanks; sorry, I forgot to include the condition of neither of the topologies being neither discrete nor indiscrete.

Bacle2 said:
Thanks; sorry, I forgot to include the condition of neither of the topologies being neither discrete nor indiscrete.
That's still not a problem. Take my X above and let ##X' = X \sqcup \mathbb R## (disjoint union; here ##\mathbb R## is given its usual topology) and map it onto ##\{\infty\} \sqcup \mathbb R## by sending X to ##\infty## and ##\mathbb R## to ##\mathbb R## via the identity.

Just wanted to tell others that I am planning to delete my posts; I realized that I phrased my OP poorly, and that this was not the question I had in mind; I didn't want to leave you hanging.

Well, the 'edit' function seems disabled, and I can't seem to be able to delete my original post. Any idea on what I could do to delete it? It was just not the question I wanted to ask.

Bacle2 said:
Well, the 'edit' function seems disabled, and I can't seem to be able to delete my original post. Any idea on what I could do to delete it? It was just not the question I wanted to ask.

Can't you just ask the new question?? Or make a new thread??

Yeah, no problem. I just thought the question, as (poorly) posed by me, is trite and not very helpful, and maybe it is better to ask more interesting questions.

## 1. What is the cardinality of the preimage f-1(y) of f:X->Y continuous?

The cardinality of the preimage f-1(y) of f:X->Y continuous is the number of elements in the set of all points in X that map to the specific value y in Y. In other words, it is the size or count of the set of all possible inputs that result in the output y.

## 2. How is the cardinality of the preimage affected by the continuity of the function f:X->Y?

The continuity of the function f:X->Y does not directly affect the cardinality of the preimage. However, a continuous function ensures that the preimage f-1(y) is also a continuous set of points, which can make it easier to determine its cardinality.

## 3. Can the cardinality of the preimage be infinite?

Yes, the cardinality of the preimage can be infinite if the function f:X->Y is a surjective (onto) function. This means that for every possible output y in Y, there is at least one input x in X that maps to it. In this case, the preimage f-1(y) would contain an infinite number of elements.

## 4. How can the cardinality of the preimage be calculated?

The cardinality of the preimage can be calculated by finding all the possible inputs in X that map to the specific value y in Y. This can be done by graphing the function, using algebraic methods, or by looking at the properties of the function (e.g. injectivity or surjectivity).

## 5. Does the cardinality of the preimage have any practical applications?

Yes, the cardinality of the preimage has several practical applications in fields such as mathematics, computer science, and statistics. It can be used to study the behavior of functions, analyze data sets, and make predictions in various real-world problems.

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