Question

define upper limit Topology ​

Answer 1

Answer:

The Upper Limit Topology on the set of real numbers , is the topology generated by all unions of intervals of the form $\{ (a, b] : a, b \in \mathbb{R}, a \leq b \}$.

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Answer 2

Answer:

The Lower and Upper Limit Topologies on the Real Numbers

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Table of Contents

The Lower and Upper Limit Topologies on the Real Numbers

The Lower Limit Topology on R

The Upper Limit Topology on R

The Lower and Upper Limit Topologies on the Real Numbers

Recall from the Generating Topologies from a Collection of Subsets of a Set page that if X is a set and B is a collection of subsets of X then B is a base for some topology τ if and only if the following two conditions are satisfed:

X=⋃B∈BB.

For all B1,B2∈B and for all x∈B1∩B2 there exists a B∈B such that x∈B⊆B1∩B2.

If the collection B satisfies the two conditions above then the generated topology is:

(1)

τ=⎧⎩⎨U:U=⋃B∈B∗B,B∗⊆B⎫⎭⎬=⎧⎩⎨⋃B∈B∗B:B∗⊆B⎫⎭⎬

We will now use this theorem to define two very important topologies on the set of real numbers R.

The Lower Limit Topology on R

Definition: The Lower Limit Topology on the set of real numbers R, τ is the topology generated by all unions of intervals of the form {[a,b):a,b∈R,a≤b}.

Another name for the Lower Limit Topology is the Sorgenfrey Line.

Let's prove that (R,τ) is indeed a topological space.

If τ is generated by unions of all intervals contained in B={[a,b):a,b∈R,a≤b} then B is a base for τ.

We will show that the conditions from the above page are satisfied to verify that τ is indeed a topology.

Clearly the first condition is satisfied since:

(2)

X=⋃a,b∈R[a,b)

For the second condition, let B1=[a,b),B2=[c,d)∈B. It is easier to visualize B1∩B2 with the following diagram:

Screen%20Shot%202015-09-24%20at%2010.54.13%20PM.png

In the first case above, we can easily find the interval B=[a,b)=[c,d) such that for all x∈B1∩B2 we have that x∈B⊆B1∩B2.

For the second case above, we can also easily find e,f∈R such that a<e<b, c<f<d such that B=[e,f) and for which every x∈B⊆B1∩B2.

For the third and fourth cases above, we see that B1∩B2=∅.

Therefore τ is generated by B and is hence a topology on R.

The Upper Limit Topology on R

Of course, we can also define the upper limit topology on R analogously as follows:

Definition: The Upper Limit Topology on the set of real numbers R, τ is the topology generated by all unions of intervals of the form {(a,b]:a,b∈R,a≤b}.

In this case, τ is generated by B={(a,b]:a,b∈R,a≤b} and verifying that τ is indeed a topology follows similarly from above.

Step-by-step explanation:

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