Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNormality versus paracompactness in locally compact spaces
84
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on omega_1, as well as of a strong form of Changs Conjecture. Together with other improvements, this enables the characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of omega_1.
rate research
Read More
We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.
A theorem by Norman L. Noble from 1970 asserts that every product of completely regular, locally pseudo-compact k_R-spaces is a k_R-space. As a consequence, all direct products of locally compact Hausdorff spaces are k_R-spaces. We provide a streamlined proof for this fact.
We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are Lindelof.
A locally convex space (lcs) $E$ is said to have an $omega^{omega}$-base if $E$ has a neighborhood base ${U_{alpha}:alphainomega^omega}$ at zero such that $U_{beta}subseteq U_{alpha}$ for all $alphaleqbeta$. The class of lcs with an $omega^{omega}$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Frechet lcs (hence spaces of distributions $D(Omega)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $omega^{omega}$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $omega^{omega}$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $varphi$ endowed with the finest locally convex topology has an $omega^omega$-base but contains no infinite-dimensional compact subsets. It turns out that $varphi$ is a unique infinite-dimensional locally convex space which is a $k_{mathbb{R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_{p}(X)$ are provided.
In this paper, we characterize stratifiable (or semi-stratifiable) spaces, and monotonically countably paracompact (or monotonically countably metacompact) spaces by expansions of locally upper bounded semi-continuous poset-valued maps. These extend earlier results for real-valued Locally bounded functions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
