Open map

A function $ f: X \to Y $ between two topological spaces is called an open map if the image of every open set in $ X $ is also an open set in $ Y $.

An open map preserves the openness of sets. In other words, if you start with an open set in the domain, its image will also be open in the codomain.

The concept of an open map, also called open function, is crucial when studying the topological properties of spaces because it provides insight into the relationship between $ X $ and $ Y $.

For example, it helps describe how the topological structure of $ X $ is transformed when mapped into $ Y $.

There is a similar concept known as a closed map. A function $ f: X \to Y $ is called closed if the image of every closed set in $ X $ is also a closed set in $ Y $. In other words, it maps closed sets to closed sets.

An example of an open map
Distinguishing open maps from continuous functions

An example of an open map

Consider the topological space $ X = \mathbb{R} $ and the identity function $ f: X \to X $, defined by $ f(x) = x $ for all $ x \in X $.

$$ f(x) = x $$

This function is always an open map because the image of any open set $ U \subset X $ is simply the set itself $ U $, which remains open in $ X $.

For instance, if we take the open interval $ (1,4) $ in the domain $ X $, its image under $ f $ will still be the open interval $ (1,4) $ in the codomain.

Distinguishing open maps from continuous functions

While continuous functions deal with the behavior of preimages (sets that are "pulled back" by the function), open maps are concerned with images (sets that are "pushed forward" by the function).

A continuous function $ f: X \to Y $ ensures that the preimage of any open set in $ Y $ is an open set in $ X $. In other words, continuity describes how open sets are "pulled back" from the codomain to the domain.
An open map, on the other hand, focuses on how open sets are "pushed forward" from the domain to the codomain. If the image of an open set in $ X $ is always open in $ Y $, then the function is open.

Thus, open maps and continuous functions are not the same!

A continuous function is not necessarily an open map.

Example

Consider the function $ f(x) = x^2 $ defined on $ \mathbb{R} $.

$$ f(x) = x^2 $$

This function is continuous but not an open map.

For example, if we take the open set $ (-2, 2) $ in $ \mathbb{R} $, the image of this set under $ f(x) = x^2 $ is $ [0, 4) $, which is not open in $ \mathbb{R} $ because it lacks an open neighborhood around $ 0 $, as $ 0 $ is a closed boundary point.

This example clearly shows that continuity does not imply that open sets are mapped to open sets.

In summary, an open map doesn't need to be continuous, and vice versa, so it's important not to assume that one property implies the other.

And so on.