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Set Theory and General Topology
Published in Kenneth Kuttler, Modern Analysis, 2017
A useful construction when dealing with locally compact Hausdorff spaces is the notion of the one point compactification of the space. Suppose (X, τ) is a locally compact Hausdorff space. Then let X˜≡X∪{∞} where ∞ is just the name of some point which is not in X which we call the point at infinity. A basis for the topology τ˜ for X˜ is τ ∪ {KC where K is a compact subset of X}. The reason this is called a compactification is contained in the next lemma.
Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
Let X be a noncompact locally compact Hausdorff space with topology T $ \boldsymbol{{ \fancyscript {T}}} $ and let ∞ $ \infty $ be a point not in X. Define X∞:=X∪{∞} $ X_\infty := X\cup \{\infty \} $ and let T∞ $ \boldsymbol{{ \fancyscript {T}}}_\infty $ consist of the members of T $ \boldsymbol{{ \fancyscript {T}}} $ together with all sets of the form X∞\K $ X_\infty \setminus K $ , where K⊆X $ K\subseteq X $ is compact. It is straightforward to check that T∞ $ \boldsymbol{{ \fancyscript {T}}}_\infty $ is closed under finite intersections and arbitrary unions and hence is a topology on X∞ $ X_\infty $ . The pair (X∞,T∞) $ (X_\infty ,\boldsymbol{{ \fancyscript {T}}}_\infty ) $ is called the one-point compactification of X. This construction is useful in extending results from a compact setting to a locally compact one, as, for example, in 2, below.
Locally Starplus-Compactness in L-Topological Spaces
Published in Fuzzy Information and Engineering, 2018
The class of locally compact spaces is far more wider than the class of compact spaces. The locally compact spaces often arise in topology and applications of topology to geometry, analysis and algebra. For example, the study of locally compact abelian group forms the foundation of harmonic analysis. It is well known that every compact space is locally compact but the converse need not be true. For example, the Euclidean space is locally compact but not compact. Topological manifolds share the local properties of Euclidean space and hence are locally compact. A locally compact space can be imbedded in a compact space, which is its compactification. One of the simplest compapctification of a space is the one point compactification, wherein one simply adjoins one new point to the space. The classical example of one point compactification is the embedding of the Gaussian plane of complex numbers into the Riemann sphere. The category of locally compact spaces has been applied in almost every subdiscipline of mathematics and hence it is important to formulate an appropriate version of local compactness in the L-fuzzy setting.
On irreducibility of oseledets subspaces
Published in Dynamical Systems, 2018
Christopher Bose, Joseph Horan, Anthony Quas
Step 1: Let denote the complex Grassmannian of one-dimensional subspaces of . It is well known that is homeomorphic to , the one-point compactification of . In particular, if we choose then a homeomorphism is given by for , and ; this yields coordinates for the Grassmannian.
Limsup is needed in the definitions of topological entropy via spanning or separation numbers
Published in Dynamical Systems, 2020
To show compactness of , it suffices to notice that for every n the restriction of to the subspaces is , and every open neighbourhood of contains all but finitely many of the spaces . Thus topologically, is the one-point compactification of .