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Every topological space X has the initial topology (or weak topology) with respect to the family of continuous functions from X to the Sierpiński space. (see http://en.wikipedia.org/wiki/Initial_topology.)

This is the two point space {a,b} with open sets: emptyset, {a} and {a,b} only. If U is any subset of X, then the function fU, mapping points in U to a and the rest to be b, will be continuous if and only if U is open.

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Every topological space X has the initial topology (or weak topology) with respect to the family of continuous functions from X to the Sierpiński space. (see http://en.wikipedia.org/wiki/Initial_topology.)

This is the two point space {a,b} with open sets: emptyset, {a} and {a,b} only. If U is any subset of X, then the function fU mapping points in U to a and the rest to be will be continuous if and only if U is open.

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Every topological space X has the initial topology (or weak topology) with respect to the family of continuous functions from X to the Sierpiński space. (see http://en.wikipedia.org/wiki/Initial_topology.)