Polish spaces Deﬁnition 2.1: A Polish space is a separable topological space X for which exists a compatible metric d such that (X,d) is a complete metric space.

$\endgroup$ – riemann Jul 4 '19 at 10:52. add a comment | 3 Answers Active Oldest Votes. ISOMETRY GROUPS OF SEPARABLE METRIC SPACES MACIEJ MALICKI AND SLAˆ WOMIR SOLECKI Abstract. Deﬁnition. ALC:almost locally compact space ; LC: locally compact space. An equivalent deﬁnition: X is locally compact at x ∈ X iﬀ there is a closed set C ⊂ x such that C contains soe neighborhood of x. Lemma 2.

A topological space X is called locally compact at a point x ∈ X if there exists a neighborhood U x 3 x such that U x is compact. Proof: is a subspace of a compact Hausdorff space : This follows from fact (1); we can take to be the one-point compactification of . Introduction Let Ck(X) be the space of continuous real-valued functions on X with the compact-open topology. 2 Almost locally compact spaces In this section we introduce the concept of an almost locally compact space .We will show that this property is preserved under an almost continuous map and is a topological property.

Information and translations of Locally compact space in the most comprehensive dictionary definitions resource on the web.

A space X is locally compact if for each x ∈ X, there exists an open neighborhood U of x with closure U¯ compact. Given: A locally compact Hausdorff space .

When X is also Hausdorﬀ, the property of local compactness becomes much stronger. What does Locally compact space mean? As mentioned before, there may be many different compatible metrics that make X complete. X is called locally compact if X is compact at every point. If, in addition, fn are holomorphic, then the limit is a normal space: This follows from fact (2). Theorem 1. De nition 2.1 . Meaning of Locally compact space. If X is already given as a complete metric space with countable dense subset, then we call X a Polish metric space. To prove: is a completely regular space. In the case that X is a non-locally compact Polish space whose derived set is compact, we show that all spaces Ck(X,2) are homeomorphic, having the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n + 1)st. 1. $\begingroup$ Does Locally Compact Seperable $\Rightarrow$ Polish?

Deﬁnition 1. Let’s state this as a theorem. Definition of Locally compact space in the Definitions.net dictionary. We show that every locally compact Polish group is iso- Find link is a tool written by Edward Betts.. searching for Locally compact space 15 found (41 total) alternate case: locally compact space Equicontinuity (1,962 words) exact match in snippet view article find links to article convergent sequence of continuous functions fn on either metric space or locally compact space is continuous. A space X is almost locally compact (denoted by ALC) at p 2 X if given a reg- ular open neighborhood …