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There are several ways I think of expressing this 'duality'. But before describing this, maybe it would help to explain a sense in which 'existence' (at least one element, or totality of a relation) i …

For a variety of reasons, it's often useful to develop an intuition for finite topological spaces. Since the only Hausdorff finite spaces are discrete, one will have to deal with the non-Hausdorff cas …

No, a path-connected Hausdorff space is arc-connected, whence it would be of (at least) continuum cardinality provided it has more than one point. This follows from a more general (and deep) result th …

Here is a method for manufacturing such topological rings. The main technical ingredient is a product-preserving functor $$\Theta: \mathrm{Set}^{\Delta^{op}} \to \mathrm{CGHaus}$$ from the categ …

Of course, Zhen has answered the question correctly under the reasonable assumption that what is being asked is whether the usual forgetful functor $U: \mathrm{Tych} \to \mathrm{Set}$ is monadic (stri …

The question doesn't seem to be very well expressed, but the intended question might be as follows. Take $\mathbb{R}^\infty$ to mean the vector space consisting of real tuples $(v_1, v_2, v_3, \ldots) …

There aren't any. Hausdorff spaces are sober spaces. If $X, Y$ are sober, then every frame map $\mathcal{O}(Y) \to \mathcal{O}(X)$, i.e., every poset map between their topologies that preserves finite …

I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hier …

To me, the second definition of local compactness is much to be preferred for the simple reason that such locally compact spaces $X$ are exponentiable in $Top$, meaning that $X \times -: Top \to Top$ …

I would motivate them as follows: if topological spaces were invented to give a general meaning to "continuous function", then uniform spaces were invented to give a general meaning to "uniformly cont …

It's called the kernel or kernel pair of $f$. It is used all over the place in category theory, for example to describe the useful notion of regular category where one sets up Galois connections which …

No, it does not. If it did, then $\mathbf{Comp}$ would be a reflective subcategory of the total category $\mathbf{Top}$, and hence would be total itself. Now, total categories admit all small limits, …

A very closely related question (and maybe the one you meant to ask?) is: which spaces $Y$ in the category of topological spaces and continuous maps are exponentiable, i.e., for which $Y$ does the fun …

The space of nowhere differentiable continuous functions $f: [0, 1] \to \mathbb{R}$, as a subspace of the space of all continuous functions under the sup-norm topology.

After seeing wood's last comment (comment #2 under his question), I've decided to add a few words (a bit too many for a comment) which hopefully make clear the force of Qiaochu's answer. Generally s …