Local Connection in Locally Compact Spaces
نویسنده
چکیده
It was proved by Hurewicz1 that a compact space which is both LC1 and lc" is LCn. In the present paper the corresponding result for locally compact spaces is proved, (a) for uniform local connection, and (b) for relative local connection.2 The extension of Hurewicz's theorem to locally compact spaces is included in (b). The main difficulty in extending Hurewicz's methods is that his "Satz 6," on the passage from e-homotopy to true homotopy, cannot be carried over to locally compact spaces without substantial modification, even when uniform local connection is assumed. To overcome this a stronger form of the lcp and LO conditions is used, namely (for lcp), the existence of a function f (5, x) such that, given a compact set F in the neighbourhood U(x, f (5, x)) of any point x, there is a compact subset F' of U(x, 8) such that every g-cycle in F bounds in F', for Ogqgp; and analogously for LO. It is shown that these are equivalent to the ordinary lcp and LO properties in locally compact (metric) spaces.
منابع مشابه
Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups
We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...
متن کاملON LOCAL HUDETZ g-ENTROPY
In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.
متن کاملA remark on Remainders of homogeneous spaces in some compactifications
We prove that a remainder $Y$ of a non-locally compact rectifiable space $X$ is locally a $p$-space if and only if either $X$ is a Lindel"{o}f $p$-space or $X$ is $sigma$-compact, which improves two results by Arhangel'skii. We also show that if a non-locally compact rectifiable space $X$ that is locally paracompact has a remainder $Y$ which has locally a $G_{delta}$-diagonal, then...
متن کاملOne-point extensions of locally compact paracompact spaces
A space $Y$ is called an {em extension} of a space $X$, if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {em equivalent}, if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Yleq Y'$, if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ ...
متن کاملSome relations between $L^p$-spaces on locally compact group $G$ and double coset $Ksetminus G/H$
Let $H$ and $K$ be compact subgroups of locally compact group $G$. By considering the double coset space $Ksetminus G/H$, which equipped with an $N$-strongly quasi invariant measure $mu$, for $1leq pleq +infty$, we make a norm decreasing linear map from $L^p(G)$ onto $L^p(Ksetminus G/H,mu)$ and demonstrate that it may be identified with a quotient space of $L^p(G)$. In addition, we illustrate t...
متن کامل