Search Results

Let $X$ be the rationals with their subspace topology, and $X^+=X\cup\{\infty\}$ be its one-point compactification. Because $X$ is not locally compact, $X^+$ is not $T_2$. The space $X^+$ has the prop …

Take $X$ to be an RC space which isn't $T_2$ such as the one-point compactification of the rationals. We will show $X^2$ is not RC. Note that it is not $T_2$ as its factors are not $T_2$. First we wil …

First let's identify the semi-open sets. We first note all open sets are open. A simpler definition of the open sets are (assuming $p=0$), $$\tau=\{U\subseteq\mathbb R:0\in U\Rightarrow\mathbb R\setmi …

The closed topologist's sine curve is the classic example of a connected but not locally connected space. But local neighborhoods away from the y-axis are copies of $\mathbb R$, which is connected. To …

KP answered the question, and in fact provided the answer to a stronger question: his space is strongly KC, showing that Hausdorff is quite necessary to show extremally disconnected spaces are both to …

Questions like these are often answerable by a search of the pi-Base (noting that every Hausdorff paracompact space is completely uniformizable): https://topology.pi-base.org/spaces?q=%20hausdorff%2B% …

EDIT: This answer relied on an accepted answer elsewhere that has now been updated to remove an oversight. See my note below. First I need to prove that the Arens-Fort space $X$ is not compactly gener …

Questions like these are easily answered with a search of pi-Base: π-Base, Search for $k_3$-space+Paracompact+~$T_2$ Six counterexamples are listed there today, including Tyrone's example. I'll sugges …

https://www.researchgate.net/publication/281110530_Destruction_of_metrizability_in_generalized_inverse_limits We worked out the details to get what we needed in that paper. Specifically, if $f$ is an …

The reference is here, provided you cannot find anything more classical. Every closed subset of a Menger set is Menger. Thus if $A(X)$ is Menger, then its closed subset $X\times\{0\}\cong X$ is Menger …

Let $X$ be a set, and let $\kappa$ be an infinite cardinal. Say sets in $X$ are closed if their cardinality is at most $\kappa$. (This class includes discrete spaces as you mentioned as well as spaces …

Writing up a direct proof for Rothberger based upon Caruvana's references. It's unclear why I didn't think to try it, but it's much easier to think about the K-Rothberger game in terms of its dual - b …

Summarizing comments as an answer. The standard interpretation of radial does not require that $\alpha\mapsto a_\alpha$ be continuous. Therefore, the Fortissimo space is radial, despite the lack of no …

Spearable implies ccc is theorem T21 of pi-Base, which references Counterexamples. The book doesn't provide a proof, but the result is standard. (Take a collection of pairwise disjoint open sets, if …

I was visiting Auburn today and obtained a scan. https://github.com/StevenClontz/research/blob/master/miscellaneous/SKM_C650i23042612550.pdf