2
votes
1answer
24 views

Cylinder with bases collapsed to a point.

The problem, although arising from some deeper facts, is quite simple. I would like to visualise the quotient space $A$ given by the cylinder $I\times S^{1}$ ($S^{1}$ is the circle in $\mathbb{R}^{2}$ ...
1
vote
1answer
49 views

Is the category of these particularly nice spaces cartesian closed?

Is the category of Hausdorff, compactly generated, locally path-connected, semi-locally 1-connected spaces (and continuous maps between them) cartesian closed? If not, in what ways does it fail to be? ...
2
votes
1answer
61 views

Topological construct

I just started working with some category theory and I would like to understand the link between what I am studying now and what I know about topological spaces. By definition, a construct (in our ...
2
votes
2answers
51 views

Derivatives on Functors

I'm not even sure if this question makes pedantic sense but is there any way to rigorously define the notion of taking the derivative of a functor?
2
votes
0answers
45 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
1
vote
1answer
70 views

Quotient Map vs Embedding (Topology)

Problem 1: Can any quotient $\tilde{X}$ of $X$ be embedded in $X$? Moreover, does any (surjective) quotient map $\pi:X\to\tilde{X}$ left split with an (injective) embedding $\iota:\tilde{X}\to X$? ...
1
vote
2answers
85 views

Proof: Categorical Product = Topological Product

Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces $X_i$ and any topological ...
3
votes
3answers
173 views

What does Structure-Preserving mean?

A very basic definition in category theory is the definition of morphism between objects. If the category is a construct, i.e., a category $\mathcal C$ equipped with a faithful functor $U\colon ...
0
votes
1answer
42 views

Characteristic Property = Universal Property?

Problem They seem to be the same -almost! But are they really or is it just unlucky accident that they look so similar however describe totally different notions? Example I was trying to set the ...
1
vote
3answers
89 views

Quotient Topology = Coproduct

Quotient topology seems to satisfy the universal property for coproducts at first glance. However, at second glance they seem to fail to fit into that frame in general since not every map passes to ...
2
votes
2answers
75 views

Algebraic Object in Topology

Common algebraic categories are group, ring, module and algebra. Some of them have the corresponding object in topology, like topological group and topological linear space. We define them by making ...
5
votes
1answer
114 views

Two topologies over the same set which are carried to each other by a lattice isomorphism induce homeomorphic spaces?

This is a question that I've been turning in my head and haven't been able to come up with a way to proceed to either prove it or construct a counter-example: Let $X$ be a set and $\mathfrak{T}_X$ ...
6
votes
3answers
173 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
2
votes
3answers
114 views

Inverse limit of an inverse system of topological spaces

Given an inverse system $\mathcal G=\{X_i\}$ of topological spaces over some directed set $I$. If $X=\prod\limits_{i\in I}X_i$, the inverse limit $X^*=\varprojlim X_i$ of $\mathcal G$ is a subspace ...
3
votes
2answers
84 views

Topology inducing Order

We know by Munkres that any (total) ordering induces a topology and (luckily) for the real line that coincides with the euclidean topology. In fact, this construction can be carried over to any ...
3
votes
3answers
80 views

Limits of Topological Vector Spaces

Let $X, Y_1, Y_2, \cdots$ be a sequence of topological vector spaces, and let $f_n : X \to Y_n$ be a sequence of continuous linear maps. Define the product space $\mathcal Y_N := Y_1 \times \cdots ...
3
votes
0answers
47 views

The compact-open topology as an initial topology

Given $X,Y \in Top$, the compact-open topology on $Y^X$ has a sub-basis consisting of sets $V_{K,U}:=\{f\in Y^X: f(K) \subseteq U\}$, for open sets $K\subseteq X$ and compact sets $U \subseteq Y$. ...
4
votes
1answer
56 views

The functor $\mathrm{Haus}\to\mathrm{Set}$ sending a space to its set of open sets is not representable?

I know the contravariant functor $\mathrm{Top}\to\mathrm{Set}$ sending a topological space to its set of open sets is representable, with representing object being the two point space with precisely ...
2
votes
1answer
33 views

“Topologification” of a subcollection of a power set

Let $X$ be any set and consider any $\mathscr{S} \subseteq \mathcal{P}(X)$, where the latter is the power set. It is natural to ask if we make $X$ a topological space by the subcollection ...
2
votes
2answers
35 views

Is there a simple way of visualising the direct limit of the cyclic subgroups of a group?

By way of background to this question, I am interested in the properties of direct limits. They are usually defined in terms that assume there is an underlying directed poset, but according to ...
3
votes
0answers
46 views

Final Topology as Colimit

My professor recently said that the final topology induced by a family of functions can be thought of as a colimit. Note that I am aware of the normal categorical description of the final topology ...
0
votes
1answer
30 views

Problem from Ravi Vakil's AG notes

Exercise 1.3.O is the following: If $X$ is a topological space, show that fibered products always exist in the category of open sets of $X$, by describing what a fibered product is. (Hint: it has a ...
7
votes
0answers
69 views

Flabby sheaf comonad

If one Googles sufficiently hard one finds the statement that Roger Godement gives the first example of a comonad, used to compute flabby resolutions of sheaves, in his monograph "Topologie alg├ębrique ...
2
votes
0answers
25 views

Disjoint Union Topology universal property as instance of final topology's universal property [duplicate]

I'm studying the final topology on a set $X$ induced by a family $\{f_\alpha:X_\alpha\rightarrow X\mid \alpha\in A\}$ and know of its universal property, namely that given a function $g:X\rightarrow ...
1
vote
0answers
72 views

What are epimorphisms in the category of Hausdorff spaces? [duplicate]

Let Haus be the category of Hausdorff spaces whose morphisms are continous maps. It seems that epimorphisms of Haus are those maps whose images are dense in the target spaces. How do you prove or ...
3
votes
3answers
135 views

Understanding the Category of open subsets of a top. space X $Op_X$

I have a problem very similar to the one posted here by Brian. I guess he also stumbled across the introduction of the category of open subsets of a topological space X (denoted by $Op_X$) in Pierre ...
4
votes
1answer
76 views

Direct limit of topoloical spaces

Let $X$ be a topological space. Suppose $X_n$ are subspaces of X with $X_1 \subset X_2 \subset ... \subset X$. I'm going to prove $\varinjlim X_n =\cup_n X_n$. I have some trouble in proving that ...
2
votes
1answer
65 views

Understanding Pushouts in Top.

The Pushout of $X \leftarrow Z\rightarrow Y$ with $f:Z\rightarrow X$ and $g:Z\rightarrow Y$ in $\mathbf{Top}$ exists and is given by $X\coprod Y/\sim,$ where "$\sim$ is the equivalence relation ...
2
votes
1answer
67 views

Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes

What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open ...
1
vote
1answer
66 views

what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...
3
votes
1answer
99 views

Category of topological pairs

Is there a standard abbreviation for the category of topological pairs? I have searched for it in vain.
3
votes
2answers
75 views

Do pushouts of compactly generated Hausdorff spaces exist?

Let $A\to X$ and $A\to Y$ be maps of compactly generated Hausdorff spaces. Does the pushout $X\coprod_A Y$ in the category of compactly generated Hausdorff spaces exist? If necessary, one can assume ...
2
votes
0answers
56 views

Is the category of topological spaces coregular?

Everything is in the title. The category Top of topological spaces and continuous mappings is not regular, but is it coregular ? Furthermore, Top isn't cartesian closed, but does it satisfy the dual ...
2
votes
1answer
68 views

Right adjoint of forgetful functor from Top

How to prove this? The forgetful functor $U:\mathbf{Top}\to\mathbf{Set}$ has a right adjoint, namely the functor $\mathbf{Set}\to\mathbf{Top}$ which equips a set with the indiscrete topology and left ...
5
votes
2answers
107 views

How to construct co-equalizers in $\mathbf{Top}$?

How to construct co-equalizers in the category $\mathbf{Top}$? Well, do co-equalizers in $\mathbf{Top}$ exist at all?
2
votes
2answers
76 views

On equalizers in Top

Wikipedia says "The equalizer of a pair of morphisms is given by placing the subspace topology on the set-theoretic equalizer." for the category $\mathbf{Top}$. What is the simplest way to prove ...
1
vote
0answers
63 views

What is the use of locally connected spaces?

One of the main properties of locally connected spaces is that their connected components are clopen and thus, they are homeomorphic to the colimit of their connected components. This is good to ...
5
votes
1answer
83 views

Does the category of Simplicial Complexes have finite limits and colimits?

Does the category of Simplicial Complexes have finite limits and colimits? Does the geometric realization functor preserve them? Thanks!
2
votes
1answer
103 views

Sophism: Discrete topology is a product topology

Let $X_1$, $X_2$ be topological spaces. Let $f_1:Y\rightarrow X_1$ and $f_2:Y\rightarrow X_2$. I will construct a (fake) product $X_1\times X_2$. Let $\pi_1$, $\pi_2$ be cartesian projections. Let ...
8
votes
3answers
195 views

Construction of a Hausdorff space from a topological space

Let $X$ be a topological space. Is there a Hausdorff space $HX$ and a continuous function $i:X\rightarrow HX$ such that for any Hausdorff space $A$ and a continuous function $j:X\rightarrow A$, there ...
7
votes
0answers
164 views

Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
5
votes
1answer
218 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
6
votes
1answer
163 views

Metric vs metrizable spaces

A. Helemskii in the book "Lectures on functional analysis" write (in my horrible translation): The category of Hausdorff topological spaces (morphisms are continuous maps) contain the full ...
4
votes
2answers
93 views

In topology $X$ is also $Y$ means homeomorphic?

E.g. $\mathbb{R}P^n$ is also the quotient space $S^n / (v \sim -v)$. And when is it safe to refer to a space as one of it's homeomorphic spaces and perform further deductions from that homeomorphic ...
28
votes
7answers
886 views

Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed ...
5
votes
1answer
209 views

Quantifiers as Adjoints in Generalized Logics

It is a well known fact that the classical universal and existential quantifiers can be seen as adjoints in certain categories. In the continuous model theory of metric structures (see ...
2
votes
1answer
63 views

How to prove that initial arrows in Haus coincide with topological embeddings?

In Joy of Cats it is stated that in category $\textbf{Haus}$ initial arrows coincide with topological embeddings (pg 135). This can be proved by showing that initial arrows in $\textbf{Haus}$ are ...
5
votes
2answers
85 views

domain of initial $f : X \rightarrow Y$ in Haus equipped with coarsest topology?

If $f:X\rightarrow Y$ is initial in category Top then it is easy to proof that (!) the topology on $X$ is the set of preimages of open sets in $Y$. Just construct topology $Z$ having the same ...
12
votes
3answers
226 views

Is there a suitable definition in categories for a closed continuous function in topology?

Working in the category of topological spaces is it possible to give a 'categorical' definition for 'a closed continuous function'? I mean something like: 'a closed continuous function' is an arrow in ...
7
votes
1answer
206 views

What distinguishes topological spaces from graphs?

Topology would not "work" if one reverted the "direction" in the definition of continuous maps $f$: $$\text{open}(x) \rightarrow \text{open}(f(x))$$ It has to be $$\text{open}(f(x)) \rightarrow ...