Assume that X is a locally compact Hausdorff space. How can the "Lindelof" property of X be interpreted, in algebraic language, for C_{0}(X)?
In particular what would be a "Lindelofization" processes and its algebraic (or NC) counterpart? May be ithe algebraic analogy would be weaker than unitization..?
The concept Lindelofization: can be considered as a natural generalization of "compactification":
For a locally Lindelof Hausdorff space X, one say the Lindelof space Y is a Lindelofization of X if X is dense in Y.
A closed subspace of a Lindelof space is Lindelof. This means that you have the same situation as by the "Alexandrov compactification". Consider a space wich is locally Lindelof but not Lindelof. You add a new point a to the space, and you define the neighborhoods of a to be the complements of closed Lindelof subspaces, (every one together with a). The new space is Lindelof: From every covering, one consider an open set containing a. Its complement is closed and Lindelof, so we can extract a countable family of open sets covering the complement. We add the open set considered, that covers a, and we found a countable covering for the new space. This seems to be an Alexandrov Lindelofication!
it is interesting question.
A topological space X has a Hausdorff one point compactification iff X is locally compact.
But Compared with the above description, I didn't see Any think about the existence of a space with Lindeloff Property except some situation that I Shall describe it as follows:
At first let me define the lindeloffication of a topological space:
Definition: A Lindeloff space Y is a one point lindeloffication of X, if X is a dense subspace of Yand |Y−X|=1
We Know that any Discrete space X has a Hausdorff One point lindelofficatin which defined as follows:
→ Add a point p to X and consider the set Y=X∪{p}. then all points of X are open and the Neighborhoods of p is of the form: U∪{p} where |X−U|≤ℵ0.
Now my Questions are as follows:
Q1: For What condition on the space X, It has a Hausdorff One point lindeloffication?
Q2: Is there an obvious example of space X which is non discrete and has a one point lindeloffication?
Compactness implies that having one point compactification is functorial or unique up to Homeomorphism. Let me recall the following theorem:
Theorem1:For X the following Are equivalent:
X is maximal compact.
Every compact subset of X is closed.
Any continuous bijection f from a compact space Y onto X is a homeomorphism.
For lindelof condition we have the same theorem:
Theorem2:For X the following Are equivalent:
X is maximal Lindelof.
The set of all closed subsets of X coincides with the set of all Lindelof subspaces ofX.
If Y is a lindelof space and f is a continuous bijection from Y onto X, Then f is a Homeomorphism.
For the sake of theorem 2 We can find that the one point compactification of an uncountable set is not maximal lindelof.
But We could fix the uniqueness in Question with the maximal lindelofness property as follows:
Q3: For which topological space, we have a maximal Hausdorff one point lindelofication.
Ackemann developed the theory of open projections in operator algebras; see for instance recent papers of Blecher for more information. I would then define non-commutative Lindelof property in the following way: Given a bounded net of open projections (p_\alpha) in A**, there is a countable subnet (p'_n) with sup_\alpha p_alpha = sup_n p'_n.
I guess it is quite natural.
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