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Professor emeritus at Cleveland State University. I’m a set-theoretic and general topologist with an interest in combinatorics. I’m also interested in linguistics, especially historical linguistics.


Jun
23
answered Ceiling function between topological spaces
Jun
23
answered Question regarding connected $T_{3}$ space
Jun
23
comment Question regarding connected $T_{3}$ space
@Stefan: A countable space is Lindelöf, and a regular Lindelöf space is normal.
Jun
23
answered Hilbert spaces, convergent sequence
Jun
23
comment Bounded non-convergent sequence with respect to an ultrafilter
@Martin: You’re very welcome.
Jun
23
answered distinguishable and indistinguishable people and ticket offices
Jun
23
revised distinguishable and indistinguishable people and ticket offices
Produced the intended formatting.
Jun
23
revised Do these two definitions of Disconnectedness coincide?
Typo.
Jun
23
comment Bounded non-convergent sequence with respect to an ultrafilter
@Martin: In the general case $\mathscr{U}$-boundedness of $\langle x_n:n\in\Bbb N\rangle$ just means that there is a $U\in\mathscr{U}$ such that $\operatorname{cl}_X\{x_n:n\in U\}$ is compact. In a compact space all sequences are therefore $\mathscr{U}$-bounded. We can simplify this to boundedness in the case of $\Bbb R$, thanks to the Heine-Borel theorem, but it’s compactness, not boundedness, that really matters. (Though since the one-point compactification of $\Bbb R$ is homeomorphic to the circle $S^1$, you can actually use boundedness with it as well, once you embed it in $\Bbb R^2$.)
Jun
23
comment Metric Space & Cauchy Sequence
@Mathy: You’re very welcome.
Jun
23
answered Do these two definitions of Disconnectedness coincide?
Jun
23
answered Why is the inverse image of a compact set under a special sort of function compact?
Jun
23
answered Metric Space & Cauchy Sequence
Jun
23
comment Error in Fibonacci recurrence proof by induction?
In addition to having the wrong summation, you have a numerical error: $F_8=21$, not $24$.
Jun
22
comment Showing one point compactification is unique up to homeomorphism
@Serpahimz: It means that $K$ is closed in $Y$ but not in $X$. That’s possible iff $p$ is a limit point of $K$ in $X$. $V\in\tau_Y$, so there is a $W\in\tau$ such that $V=W\cap Y$. $W$ must be either $V$ or $V\cup\{p\}$, and the latter isn’t in $\tau$, so $V\in\tau$. Thus, each $y\in Y\setminus K$ has $V$ as a $\tau$-open nbhd disjoint from $K$. If $K$ is not $\tau$-closed, that leaves $p$ as the only possible member of the non-empty set $(\operatorname{cl}_XK)\setminus K$.
Jun
22
comment Metric Space & Cauchy Sequence
Most of this is pretty straightforward; have you made any progress on any part of it? Note that (2), (3), and (4) are independent of (1), that it’s entirely possible to do (3) without having done (2), and that (4) is easily done without having done any of the rest.
Jun
22
comment Showing one point compactification is unique up to homeomorphism
@Serpahimz: You’re welcome, and thank you.
Jun
22
revised Bounded non-convergent sequence with respect to an ultrafilter
added 549 characters in body
Jun
22
comment Showing one point compactification is unique up to homeomorphism
@Serpahimz: Yep; good catch. Thanks.
Jun
22
revised Showing one point compactification is unique up to homeomorphism
Typo.