When is the union of topologies a topology?
2017年4月25日 · A union of topologies is a topology 0 Defining the class of sets for which the union of topologies is a topology [sets of subsets of the set of topologies on a set: the settening]
Comparing Topologies on $\\mathbb{R}$ - Mathematics Stack …
2016年2月28日 · 1. Consider the following topologies on R : T1 = the standard topology. T2 = topology of RK. T3 = the finite complement topology. T4 = the upper limit topology, having all sets (a, b] as basis. T5 = the topology having all sets (−∞, a) as basis. RK is the topology on R with basis elements (a, b) and (a, b) − K where K ={1 n: n ∈N}.
real analysis - Equivalent Topologies - Mathematics Stack Exchange
Two topologies are called equivalent if any non-empty open set of the first topology contains a non-empty open set of the second topology and, conversely, every non-empty open set of the second topology contains non-empty open sets of the first topology. This does NOT mean that the two topologies are the same.
All topologies on $X=\{ a,b - Mathematics Stack Exchange
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general topology - describing all possible topologies on a set ...
1. The list is complete because X X has only four subsets ∅, {a}, {b}, X ∅, {a}, {b}, X after all, the subsets ∅ ∅ and X X must occur in each topology, and that leaves only two yes/no choices (and hence four possibilites) as to whether or not {a} {a} and {b} {b} are open. You listed these four combinatorical possibilities - and they all ...
Examples of topologies on R - Mathematics Stack Exchange
If so, then the topology on $\mathbb {R}$ generated by $ (a,b]$ is neither trivial, discrete, or the usual metric topology. Similarly, using $ [a,b)$ as generating sets gives another distinct and interesting topology. Another topology on $\mathbb {R}$, which I think was used when I first learned topology as the first example of a nontrivial non ...
general topology - How are a set of matrices a topological space ...
2012年10月10日 · You can take any sensible topology on $\mathrm{Mat}_{m,n}(\mathbb{R})$ and you will get the usual topology of $\mathbb{R}^{mn}$. By sensible I mean topology which makes addition of matrices and multiplication by scalars continuous (this …
Intersection of topologies - Mathematics Stack Exchange
2013年7月16日 · Condition 1) is trivially satisfied since X and empty set are in every topology they will be in the intersection as well. Condition 2) and 3) Here we need to prove that arbitrary unions and finite intersections of the elements in the intersection will be in the intersection. We explain this using intersection of two topologies over X X.
What exactly is the topology generated by the union of a family of ...
The discrete topology (all subsets open) is a topology for any set. So, take the set of all topologies containing the family of topologies (remember: the discrete one is here) and you will get a topology. It will be the smallest topology containing all topologies in the family. This is integral to the notion of weak topology.
general topology - Finite number of topologies on an infinite set ...
2016年1月29日 · On a countable set one can already put infinitely many pairwise non-homeomorphic Hausdorff topologies. Substantial additional conditions are needed to obtain a finite number of topologies on an infinite set. One thought is R.L. Moore's 1920 theorem characterizing the unit interval as a continuum (compact, connected metric space) with at least ...