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princy
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how does a lower limit topology strictly finer than a standard topology? please explain lemma 13.4 of munkres' topolgy..
Tinyboss said:Can you write a basis element of the standard topology as a union of basis elements in the lower limit topology? Hint: yes.
princy said:thanks for ur hint.. i got the idea.. am now not able to prove it for k-topology... that is.. how is it possible that the basis element of standard topology is the basis element of k-toplogy??..
it is stated that.."given a basis element of (a,b) of T and a point x of (a,b),this same interval is a basis element for T'' "...
i'm not getting it clear.. can u help me in this??
The Lower Limit Topology, also known as the Sorgenfrey line, is a topology on the real line in which the basis consists of all half-open intervals [a, b) where a < b. This topology is finer than the standard Euclidean topology, but coarser than the discrete topology.
Lemma 13.4 of Munkres' Topology is a result that states that if a subset A of the Lower Limit Topology is countable, then A has no limit points. In other words, every point of A is isolated.
Lemma 13.4 helps us understand the discrete nature of the Lower Limit Topology. It shows that any countable set in this topology will have no limit points, meaning that there is no way to approach any point in the set from other points in the topology. This is in contrast to other topologies, where limit points are essential for understanding the continuity and connectedness of the space.
No, Lemma 13.4 only applies to countable sets in the Lower Limit Topology. In fact, there are uncountable sets in this topology that do have limit points. For example, the set of all irrational numbers in the Sorgenfrey line has limit points, despite being uncountable.
The Lower Limit Topology and Lemma 13.4 are useful in understanding the behavior of certain functions, such as the derivative of a function. They also have applications in probability theory and measure theory, where they are used to construct counterexamples and explore the properties of different topologies.