http://www.mathreference.com/top,disc.html Obviously, these spaces are mostly of interest when λ is an infinite ordinal; otherwise (for finite ordinals), the order topology is simply the discrete topology . When λ = ω (the first infinite ordinal), the space [0,ω) is just N with the usual (still discrete) topology, while [0,ω] is the one-point compactification of N . See more In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally … See more Several variants of the order topology can be given: • The right order topology on X is the topology having as a base all intervals of the form $${\displaystyle (a,\infty )=\{x\in X\mid x>a\}}$$, together with the set X. • The left order … See more Ordinals as topological spaces Any ordinal number can be made into a topological space by endowing it with the order topology (since, being well-ordered, an ordinal is in … See more If Y is a subset of X, X a totally ordered set, then Y inherits a total order from X. The set Y therefore has an order topology, the induced order … See more Though the subspace topology of Y = {–1} ∪ {1/n}n∈N in the section above is shown to be not generated by the induced order on Y, it is … See more For any ordinal number λ one can consider the spaces of ordinal numbers $${\displaystyle [0,\lambda )=\{\alpha \mid \alpha <\lambda \}}$$ together with the … See more • List of topologies • Lower limit topology • Long line (topology) • Linear continuum See more
Order Topology -- from Wolfram MathWorld
Web2 The order topology on Z + is the discrete topology. The Product Topology De nition Then theproduct topologyon the cartesian product X Y is the topology generated by the … Webdiscrete) is disconnected. 9. !+ 1, ! 1 and ! 1 + 1 are all disconnected, since in each space the minimal element of the order is clopen as a singleton. More generally, any well-order with its order topology is disconnected (provided that it contains more than one point). 10. R nf0g(with its usual subspace topology) is disconnected. If you have ... problem with work or school account
Order topology - HandWiki
Webin this video i explain the discrete and indiscrete topology with a lot of examples.#topology #mathswithshubham #topologicalspaceWATCH OUR COMPLETE PLAYLIST... WebAug 12, 2016 · subspace A which has the discrete topology (under the subspace topology) must be countable. Under these conditions for X, B, and A, for each a ∈ A there is a basis ... [0,1] under the order topology induced by the dictionary order. In this topology (which is different from the subspace topology on [0,1] ×[0,1] as a subspace of R× R … Webdiscrete) is compact if and only if Xis nite, and Lindel of if and only if Xis countable. More generally, any nite topological space is compact and any countable topological space is Lindel of. 5.For any set X, (X;T indiscrete) is compact. 6.[0;1] with its usual topology is compact. This is not obvious at all, but we will prove it shortly. problem with words with friends 2