Posted by J. Holmer on September 30, 2007, 4:00 pm, in reply to "weak topologies" Regarding "But if we fix a topology for a space Y then continuity of a function f:X-->Y with respect to the stronger topology on X doesn't imply continuity with respect to the weaker topology on X?", the answer is yes, this too is correct. In fact the implication goes the other way: continuity with respect to the weaker topology on X implies continuity with respect to the stronger topology on X. If we put the discrete topology on X, then every function f:X\to Y is continuous. Both of your statements are correct and consistent. In a weaker topology, there are (possibly) more convergent sequences, making it harder for a function to be sequentially continuous.
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Regarding "So is it true that convergence of a sequence in a space X with respect to a stronger topology implies convergence in a strictly weaker one?", the answer is yes.
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