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Order topology
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=== Ordinal-indexed sequences === If ''α'' is a limit ordinal and ''X'' is a set, an ''α''-indexed sequence of elements of ''X'' merely means a function from ''α'' to ''X''. This concept, a '''transfinite sequence''' or '''ordinal-indexed sequence''', is a generalization of the concept of a [[sequence]]. An ordinary sequence corresponds to the case ''α'' = ω. If ''X'' is a topological space, we say that an ''α''-indexed sequence of elements of ''X'' ''converges'' to a limit ''x'' when it converges as a [[net (mathematics)|net]], in other words, when given any [[neighborhood (mathematics)|neighborhood]] ''U'' of ''x'' there is an ordinal ''β'' < ''α'' such that ''x''<sub>''ι''</sub> is in ''U'' for all ''ι'' ≥ ''β''. Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω<sub>1</sub> is a limit point of ω<sub>1</sub>+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω<sub>1</sub>-indexed sequence which maps any ordinal less than ω<sub>1</sub> to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω<sub>1</sub>, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable. However, ordinal-indexed sequences are not powerful enough to replace nets (or [[filter (mathematics)|filter]]s) in general: for example, on the [[Tychonoff plank]] (the product space <math>(\omega_1+1)\times(\omega+1)</math>), the corner point <math>(\omega_1,\omega)</math> is a limit point (it is in the closure) of the open subset <math>\omega_1\times\omega</math>, but it is not the limit of an ordinal-indexed sequence.
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