# Does there Exist a Continuous Map ?

• Bacle2
In summary, the conversation revolves around the question of whether there is always at least one continuous map between two given topological spaces. The suggestion is to find a map that satisfies the condition f-1(U)=V for every U in the second space's topology T'. It is proposed that for the infinite case, the choice of assigning each U in T' with a V in T would result in a well-defined map between the two spaces. This idea is also suggested for defining a measurable map between two measurable spaces. However, it is clarified that a constant map can be used if the second space is not empty. The speaker is just curious to see if the proposed setup would still work.

#### Bacle2

Hi, All:
I saw this question somewhere else: we are given any two topological spaces (X,T), (X',T'), and we want to see if there is always at least one continuous map between the two. The idea to say yes is this: we only need to find f so that f-1(U)=V , for every U in T', and some V in T. So, it seems , for the infinite case, we could use choice to assign to each U in T' some V in T. Does this give us a well-defined map between (X,T) and (X',T')? It seems like we could then , similarly, define a measurable map between two measurable spaces using the same idea. Does this work?
Thanks.

Just to clarify: I know if X' as above is not empty, then any constant map is continuous; I was just curious to see if the setup above worked.

## 1. What is a continuous map?

A continuous map is a function between two topological spaces that preserves the structure of the spaces. In other words, small changes in the input result in small changes in the output. This means that the points close to each other in the domain will be mapped to points close to each other in the range.

## 2. How do you determine if a map is continuous?

A map is continuous if for every open set in the range, its pre-image is an open set in the domain. In simpler terms, a map is continuous if the inverse image of open sets in the range are open sets in the domain.

## 3. Can a continuous map have a discontinuous inverse?

Yes, it is possible for a continuous map to have a discontinuous inverse. This is because continuity only applies to the forward direction of the map, not the inverse. However, if both the map and its inverse are continuous, then the map is said to be a homeomorphism.

## 4. What is the importance of continuous maps in mathematics?

Continuous maps are important in mathematics because they allow us to study the behavior of functions and spaces in a smooth and structured manner. They are used in many areas of mathematics, including analysis, topology, and geometry.

## 5. Does every map have a continuous extension?

No, not every map has a continuous extension. For example, the map from the real numbers to the integers that rounds down to the nearest integer cannot be extended to a continuous map. However, if the domain is compact and the range is Hausdorff, then every map has a continuous extension.