a problem on the topological properties of a annulus

Which properties hold for the following set?

Open, connected, compact, closed.

$A=f(B)\subset X$ where $B=\{(x,y) \in\mathbb{R}^2: 1\leq x^2+y^2\leq 2\}$,
$X$ is an arbitrary topological space and $f: \mathbb{R}^2\to X$ is an arbitrary continuous map.

My thoughts:
The given set $B$ is closed annulus and it is connected and also compact. Since $f$ is continuous so $A$ will compact and connected.
But how can I verify that closed /open/both/neither.

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As for counterexamples, let $X = \{a,b\}$ with the trivial topology $\{\emptyset,X\}$ and define $f:B\to X$ to be a constant $b$ map. Clearly, $f(B) = b$ which is neither open nor closed.
However, if $X$ is Hausdorff, $A$ will be a closed set by compactness. –  Olivier Bégassat Jan 2 '13 at 12:52