Is the set of conditions for Devaney's definition of chaos minimal?

Question

I am reading Devaney's definition of chaos. Which says:
Let $V$ be a set. $f:V \rightarrow V$ is said to be chaotic on V if

1. $f$ has sensitive dependence on initial conditions
2. $f$ is topologically transitive
3. periodic points are dense in $V$

It seems that conditions 2 and 3 implies 1. Also, condition 2 seems to imply 1. Am I missing something here?

Answer 1

Brooks, Cairns, Davis and Stacey proved that 2. and 3. imply 1. on metric spaces with an infinite number of points. There are no more redundancies in general metric spaces. However, if $V=[a,b]\subset\mathbb{R}$ and $f$ is continuous, then 2. implies 1. and 3. This is a result of Vellekop and Berglund, published in the American Mathematical Monthly, vol. 101, 1994.

Answer 2

over a metric space see the following

http://www2.math.uu.se/~warwick/vt04/DynSyst/reading/BanksEtAl.pdf