In mathematics, the theory of selection principles deals with the possibility of obtaining mathematically significant objects by selecting elements from sequences of sets. The studied properties mainly include covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially functions spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property.
== Background and definitions ==
In 1924, Karl Menger introduced the following basis property...
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In the definition of a manifold $M$, we have the following conditions:
For some fixed $n$, $M$ is locally homeomorphic to $\mathbb{R}^d$.
$M$ is connected, second countable, and Hausdorff.
Now, with this definition, it is a well-known theorem that $M$ can be embedded in $\mathbb{R}^{2n+1}$. I...
Let $\omega_1$ be the first uncountable ordinal. Let $L$ denote $\omega_1 \times [0,1)$ with the order topology and smallest element removed. How can we show this space is not second countable?
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After much delay, I have today removed the topology⇒general-topology tag synonym. This means that as of now topology is a perfectly acceptable stand-alone tag. Topology, of course, is also a wide ranging field of study, and I feel that the topology tag by itself often won't adequately narrow down...
