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initial source∗ kompik† 2013-03-21 20:49:06 fi Let A be a concrete category over X. A source (A → Ai )i∈I in A is called initial provided that an X-morphism f : |B| → |A| is an A-morphism whenever each composite fi ◦ f : |B| → |Ai | is an A-morphism. The dual notion is called a final sink. A source (A, fi )I in the category of topological spaces Top is initial if and only if A has the initial topology with respect to the family (fi )I . A topological space X is completely regular if and only if the source S(X, R), consisting of all continuous maps from X to the real line, is initial (in the construct Top); and X is a Tychonoff space if and only if S(X, R) is an initial mono-source. A similar characterization holds for epireflective subcategories of Top. References [1] J. Adámek, H. Herrlich, and G. Strecker. Abstract and Concrete Categories. Wiley, New York, 1990. ∗ hInitialSourcei created: h2013-03-21i by: hkompiki version: h38110i Privacy setting: h1i hDefinitioni h18A05i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1
