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Is there any space with such properties? If there is no such space, I would be very grateful if you could tell me proof of this fact.

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5  
Er...any singleton? –  DonAntonio Mar 25 '13 at 15:02

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There is a construction which associates a topological space to any poset. Under this construction finite posets gives rise to finite topological spaces. It is known that the space is contractible if the poset has maximum (or minimum). This construction gives you a lot of examples. For instance the example of Arthur Fischer correspond with the poset with 2 (comparable) points.

EDIT: It is "easy" to construct examples of simply-connected non-contractible spaces. An easy one is the space associated to the poset $P=\{(a,b): a=0,1; b=0,1,2\}$ where $(a,b)<(a',b')$ if $b<b'$ (this is a model of $S^2$).

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The Sierpiński space is contractible, hence simply connected.

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