Continuous Map (function in topology)

Definition: continuous

A function $f:X\to Y$ between two topological spaces is called continuous if for every open set $V\subseteq Y$, its pre-image $f^{-1}(V)$ is open in $X$ pre-image: the set of points in $X$ that are mapped to the set $V$

Note that $f$ does not need to be injective of surjective

For simplicity we normally calls a continuous function as a map

Property

A function $f:X\to Y$ is continuous $⟺$ each point of $X$ has a neighborhood on which $f$ is continuous

Conclusion

Constant maps are continuous (maps to a single point) Identity maps are continuous

If $f:X\to Y,g:Y\to Z$ are continuous, $g\circ f:X\to Z$ is continuous

Open Maps

A continuous map $f:X\to Y$ is called open if for every open $U\subseteq X$, its image $f(u)$ is open in $Y$

Note

The inverse definition of continuous maps

Closed Maps

A continuous map is call closed if it takes closed subsets to closed sets.

Warning

not every continuous map is a open map: $f:\mathbb{R}\to \mathbb{R}$ with $y\mapsto 0$ is a closed but not open map

Proposition

Let $f:X\to Y$ be a continuous map that is open or closed

• If $f$ is bijective, it is a homeomorphism

  • $f$ is a Homeomorphism $⟺f$ is open and closed

• If $f$ is injective, it is a topological embedding

  • a topological embedding is a map that is a homeomorphism onto its image

• If $f$ is surjective, it is a quotient map

• continuous maps preserves connectedness