Point set topology questions

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...

The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...

Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk [or, if you prefer, the closed unit ball in $R^n$] to itself. Does there always exist a point $x$ such that $f[...

Let $[X,\tau], [Y,\sigma]$ be two topological spaces. We say that a map $f: \mathcal{P}[X]\to \mathcal{P}[Y]$ between their power sets is connected if for every $S\subset X$ connected, $f[S]\subset Y$ ...

Without prethought, I mentioned in class once that the reason the symbol $\partial$ is used to represent the boundary operator in topology is that its behavior is akin to a derivative. But after ...

[This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.] Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...

This question is of course inspired by the question How to solve f[f[x]]=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f[f[x]]=g[x]$ on a ...

Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17[3] 357453? ...

In some circumstances, an injective [one-to-one] map is automatically surjective [onto]. For example, Set theory An injective map between two finite sets with the same cardinality is surjective. ...

This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ...

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...

Sometimes I see spider webs in very complex surroundings, like in the middle of twigs in a tree or in a bush. I keep thinking if you understand the spider web, you understand the space around it. ...

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ σ just in case every τ open set is open in σ. ...

There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k[X,\mathbb{Z}/2\mathbb{Z}] \to H^{k+i}[X,\mathbb{Z}/2\mathbb{Z}]$. [Coefficient group suppressed from ...

$\require{AMScd}$ Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ [monomorphisms into $X$ up to equivalence] as open. The requirements ...

Let X be a real orientable compact differentiable manifold. Is the [co]homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...

I would like some help, because I am getting mad trying to answer the following Question: Let $X$ be a topological space, what is a continuous path in $X$? Well, maybe you're already getting ...

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f[I]$ of $\mathbb{R}^2$. Is it true that there must be a ...

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon: [1] ${\cal X}$ is a parameterized family of somethings. [Varieties, schemes, manifolds, ...

From my limited perspective, it appears that the understanding of a mathematical phenomenon has usually been achieved, historically, in a continuous setting before it was fully explored in a discrete ...

In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" [sequences/...

What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?

In a recent blog post Terry Tao mentions in passing that: "Class groups...are arithmetic analogues of the [abelianised] fundamental groups in topology, with Galois groups serving as the analogue of ...

Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books [Topology and Geometry-Bredon, Complex ...

A subset of is meagre if it is a countable union of nowhere dense subsets [a set is nowhere dense if every open interval contains an open subinterval that misses the set]. Any countable set ...

This question concerns the ramifications of the following interesting problem that appeared on Ed Nelson's final exam on Functional Analysis some years ago: Exam question: Is there a metric on the ...

The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous. Or so I'm told, but this leaves me ...

Here's a problem I've found entertaining. Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups [integer coefficients] or one of its fundamental groups ...

If a metric space is separable, then any open set is a countable union of balls. Is the converse statement true? UPDATE1. It is a duplicate of the question here //math.stackexchange.com/...

Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of ...

Edit: Infos on the current state by Lieven Le Bruyn: //www.neverendingbooks.org/grothendiecks-gribouillis Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are [...

Consider a non-empty set $X$ and its complete lattice of topologies [see also this thread]. The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...

For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? [i.e. Can ...

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact. ...

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected? Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...

Background Let $[X,x]$ be a pointed topological space. Then the fundamental group $\pi_1[X,x]$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...

It is well known that most topological spaces can be studied via their algebra of continuous real-valued [or complex-valued] functions. For instance, in the setting of compact Hausdorff spaces, there ...

Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions": $\cdots$ Two naive questions from an outsider: [1] Have all $24$ now been resolved? [2]...

This question is being asked on behalf of a colleague of mine. Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...

I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ [...

In his 1967 paper A convenient category of topological spaces, Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces as a good replacement of the category Top topological ...

Schauder's Conjecture: "Every continuous function, from a nonempty compact and convex set in a [Hausdorff] topological vector space into itself, has a fixed point." [Problem 54 in The Scottish ...

I have never studied any measure theory, so apologise in advance, if my question is easy: Let $X$ be a measure space. How can I decide whether $L^2[X]$ is separable? In reality, I am interested in ...

Does $\mathbb C\mathbb P^\infty$ have a [commutative] group structure? More specifically, is it homeomorphic to $FS^2$, [the connected component of] the free commutative group on $S^2$? $\mathbb C\...

Possible Duplicate: Cohomology and fundamental classes Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi_*[N]\in H_*[M]$ ...

My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric. That is, let us define that a topological ...

Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by Loomis: //archive.org/details/introductiontoab031610mbp //ia800309.us.archive....

Here I mean the version with all but finitely many components zero.