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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

3 votes

Definition of Connected Subspace

Two sets are called disjoint if they have no element in common. Thus for two subsets of $Y\subset X$ there is no distinction to be made between "disjoint in $Y$ and "disjoint in $X$". Call two sets i …
Tom Goodwillie's user avatar
9 votes
Accepted

Does every ultrafilter has single limit imply Hausdorff separation

An ultrafilter $U$ on the space $X$ is said to converge to the point $p$ if every neighborhood of $p$ belongs to $U$. (This is equivalent to saying that the complement of an element of $U$ is never a …
Tom Goodwillie's user avatar
3 votes
Accepted

closed connected subspace of a cartesian product

Start with any example of a continuous bijection $f:A\to B$ between connected CW spaces that is not a homeomorphism. For example, $A$ a closed half-line and $B$ a circle. Let $Y$ be $A\times B$, choo …
Tom Goodwillie's user avatar
7 votes

locally connected versus locally compact

If $X$ is a space and "P" is an adjective that can apply to spaces, then in many cases "$X$ is locally P" means "for every point $x\in X$, for every neighborhood $N$ of $x$, there is a neighborhood $N …
Tom Goodwillie's user avatar
3 votes

Continuity of maps in which preimage preserves compactness

$f$ need not be continuous. Let $Y$ be a countably infinite set with the discrete topology. Let $X$ be the same set with a different Hausdorff topology. Let $f$ be the identity map.
Tom Goodwillie's user avatar
4 votes
Accepted

Continuous projective geometry on the interval

If $(P,L)$ is an abstract projective plane, then for any point $p\in P$ and any line $\ell\in L$ not incident to $p$ there is a bijection between the set of points incident to $\ell$ and the set of li …
Tom Goodwillie's user avatar
13 votes
Accepted

Is the defining bijection for a pullback of topological spaces a homeomorphism?

Yes. The functor $Top(T,-)$ preserves limits because it is a right adjoint.
Tom Goodwillie's user avatar
4 votes

When does a homeomorphism split into essentially minimal homeomorphisms?

No. Let $X$ be a closed interval and let $\phi:X\to X $ be an order-preserving bijection that fixes only the endpoints. Edit: This is an answer to a trivial question which is not the intended one.
Tom Goodwillie's user avatar
1 vote

Do Smash Products and Quotients Commute?

$X\wedge X\subset (Y\wedge X)\cup (X\wedge Y)\subset Y\wedge Y$, with quotients $((Y/X)\wedge X)\vee (X\wedge (Y/X))$ and $(Y/X)\wedge (Y/X)$.
Tom Goodwillie's user avatar
8 votes
Accepted

Weak homotopy equivalence of $H$-spaces

If $H$ and $G$ are spaces having abelian fundamental groups (for all basepoints), and if there is a natural isomorphism between the sets $[X,H]$ and $[X,G]$ for finite CW complexes $X$, then $H$ and $ …
Tom Goodwillie's user avatar
51 votes

Is there a Whitney Embedding Theorem for non-smooth manifolds?

A bit off-topic, but I'd like to mention some big differences between Whitney's $2n$ theorem and his $2n+1$ theorem. The idea of the $2n+1$ theorem is that "most" smooth maps from a compact smooth $ …
Tom Goodwillie's user avatar
20 votes
Accepted

What topological principle is at work here?

EDIT: I've added an argument at the end that proves a more general statement and does so more simply. I'll show that if $X$ is a finite simplicial complex having even Euler characteristic then there i …
Tom Goodwillie's user avatar
3 votes
Accepted

Homotopy equivalence of maps with compact support and maps which vanish at infinity

Let's assume $X$ is locally compact Hausdorff, so that $\hat X$ is compact Hausdorff and you can indeed use $C(\hat X,Y)$. If the inclusion $i:C_c(X,Y)\to C(\hat X, Y)$ is a homotopy equivalence for …
Tom Goodwillie's user avatar
7 votes

Does $\mathbb C\mathbb P^\infty$ have a group structure?

Let $G$ be a set. Let $EG$ be the realization of the nerve of the category that has $G$ as object set, with one morphism for each source and target. This is contractible if $G$ is not empty. If $G$ is …
Tom Goodwillie's user avatar
14 votes

Associativity of topological join and join of spheres

If $A$ and $B$ are subsets of $\mathbb R^n$ then you can map $A\times I\times B$ to $\mathbb R^n$ by $(a,t,b)\mapsto (1-t)a+tb$. If the resulting continuous map $A*B\to \mathbb R^n$ happens to be one …
Tom Goodwillie's user avatar

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