General topology
For any discussions about general topology. For instructions how to render MathJax(TeX) in chat see http://meta.math.stackexchange.com/a/3297 "Sets are not doors"
2d ago – Jakobian
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This proof can be considered a variation of the proof using ultrafilters on $X$. I want mainly to point out that we can avoid transferring the ultrafilters through the projections if we use a slightly more general characterization of compactness using ultrafilters.$\newcommand{\FF}{\mathcal F}\ne...
Can someone please help me understand why the underlined statement must be true. Why must there be a $d$ in $B_0$ less than $c$? Why can't $c$ be the smallest element of $B_0$?
A function $f :X \to Y$ between topological spaces $X,Y$ is defined to be
continuous if $f^{-1}(V)$ is open in $X$ for all open $V \subset Y$,
open if $f(U)$ is open in $Y$ for all open $U \subset X$,
closed if $f(C)$ is closed in $Y$ for all closed $C \subset X$.
Many questions in math.stac...In mathematics, in the field of topology, a topological space is said to be a door space if every subset is open or closed (or both). The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".
== Properties ==
Here are some facts about door spaces:
A Hausdorff door space has at most one accumulation point.
In a Hausdorff door space if
x
{\displaystyle x}
is not an accumulation point then
{
x
}
{\displaystyle \{x\}}
is open...