# A notion reverse to continuity, does it have a name?

Does the following notion (in some sense reverse to continuity) have a name? What are interesting properties of this concept?

$$f$$ is "anticontinuous" in $$x$$ if $$\forall \epsilon>0 \exists \delta>0:f[(x-\epsilon;x+\epsilon)]\supseteq (f(x)-\delta;f(x)+\delta).$$

(Here $$f[X]$$ is the image of a set $$X$$ under a map $$f$$.)

As Berci commented, when we demand that this hold for all $$x$$ in the domain of the function we're just saying that $$f$$ maps open sets to open sets - this is the definition of an open map. Similarly, a closed map is a map sending closed sets to closed sets. Closedness, openness, and continuity are fundamentally independent:

• If $$X$$ has more than one element, the identity map on $$X$$ is closed and open but not continuous when we equip the domain with the indiscrete topology and the codomain with the discrete topology.

• The "left inclusion" map $$\mathbb{R}\rightarrow\mathbb{R}^2:a\mapsto (a,0)$$ (with the usual topologies on $$\mathbb{R}$$ and $$\mathbb{R}^2$$) is closed and continuous but not open.

• The "left projection" map $$\mathbb{R}^2\rightarrow\mathbb{R}: (a,b)\mapsto a$$ is continuous and open but not closed.

It's a good exercise to check each of these examples.

(Meanwhile, the obvious fourth notion - "preimages of closed sets are closed" - is equivalent to continuity, since the complement of the preimage is the preimage of the complement).

I've seen in conversation the term "open at $$x$$" used to refer to the situation you describe, but I haven't seen that in a formal text so I don't know if it's universally accepted.