Open but not continuous

Question

In general there is no relation between the conditions of openness and continuity for functions between topological spaces.

For example, $f\colon (\mathbb{R}, \text{left ray topology})\to (\mathbb{R}, \text{discrete topology})$ given by $f(x)=x$ is open but not continuous.

My question if there is a function $(\mathbb{R}, \text{usual topology})\to (\mathbb{R}, \text{usual topology})$ which is open but not continuous?

And if so, what the conditions must be in this function to be open and not continuous?

I edited the question for formatting and grammar. I left your last sentence as is, because I have no idea what it means. — Alex Kruckman, Apr 22 '18 at 19:09

score 4 · Answer 1 · answered Apr 22 '18 at 19:02

4

The Conway base 13 function $f: \mathbb{R} \to \mathbb{R}$ is a rather extreme example: it is nowhere continuous but $f[(a,b)] = \mathbb{R}$ for all $a < b$ in $\mathbb{R}$, and this trivially implies that $f$ is open.

answered Apr 22 '18 at 19:02

Henno Brandsma

242,131

Beautiful example of the crazyness of real functions. – Surb Apr 22 '18 at 19:22

Open but not continuous

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