Peano space is a continuous image of the unit interval I A Hausdorff space $X$ is a Peano space if it is a continuous image of the unit interval $I$ 
Definition :- A Peano space is a compact, connected, locally connected metric space. 
update :-
$I$ is is a compact, connected and the continuous image of it is so .
How I can start with locally connected metric space ?
 A: This is the easy part of the Hahn-Mazurkiewicz theorem. So I had some fun tracking references inside Willard:
Suppose $f: [0,1] \to X$ is continuous and onto, where $X$ is a Hausdorff topological space.


*

*Willard Thm. 17.7 (cont. image of compact is compact) implies that $X$ is compact because $[0,1]$ is compact (example 17.9, it's closed and bounded in $\mathbb{R}$).

*26.3 (continuous image of connected is connected) implies that $X$ is connected. Example 26.2c says $[0,1]$ is connected.

*$f$ is a quotient map because it is a closed map (see proof of 17.14)

*So 27.12 then implies, because $f$ is quotient, that $X$ is locally connected.

*Corollary 23.2 (of Urysohn's metrisation theorem) says that a continuous image of a compact metric space inside a Hausdorff space is metrisable. So $X$ is metrisable.
Putting all together we see that $X$ is a Peano space.
A: Let $f$ be a continuous map of $I$ onto $X$. 
$X$ must be compact and metric.
Moreover, $X$ is the continuous image of a connected space and a quotient of a locally connected space, so $X$ has these 
properties itself.
Thus, $X$ is a Peano space.
(To see this, let $H$ be closed in $I$, then $H$ is compact, so $f(H)$ is also compact.
Now, $f(H)$ is compact in a Hausdorff space $X$, hence $f(H)$ is closed. Therefore, $f$ is a quotient map).
