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Questions tagged [gn.general-topology]

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

8
votes
3answers
718 views

Relatively countably compact subsets without countably compact closure.

I am looking for a (Hausdorff or better) space $X$ and a subset $A$ of $X$ that is relatively countably compact (every sequence from $A$ has an accumulation point in $X$) such that the closure of $A$ ...
51
votes
10answers
7k views

Non-homeomorphic spaces that have continuous bijections between them

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 $g: Y \to X$?
-1
votes
2answers
439 views

Union of proximally connected sets

Let (δ;U) is a proximity space. I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y. Is the following true? (I need a proof or a counter-example.) Conjecture If S ...
1
vote
3answers
2k views

Definition of Connected Subspace

In Munkres (Chapter 3, Section 23, p. 148), Munkres shows that if a subspace $Y$ of a space $X$ is not connected, then there are two disjoint open subsets $A,B$ such that the union of $A$ and $B$ ...
9
votes
1answer
2k views

The Wedge Sum of path connected topological spaces

A definition of wedge sum can be found here: http://en.wikipedia.org/wiki/Wedge_sum My professor has claimed that wedge sums of path connected spaces X and Y are well-defined up to homotopy ...
2
votes
1answer
2k views

What is the pure intuition for topological continuity and topology? [closed]

I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity. The ...
6
votes
3answers
509 views

profinite spaces coming from profinite groups

This is probably well-known: Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed? - Is every profinite group ...
3
votes
3answers
2k views

motivation for compactness [duplicate]

Possible Duplicate: How to understand the concept of compact space Hello, I am learning some analysis on my own and what is the motivation to consider compactness? eg. I do not understand why ...
-2
votes
1answer
473 views

Countable open subgroup

In a Hausdorff topological group, how can I show that every infinite topological group has a countable open subgroup?
3
votes
1answer
472 views

Functoriality of base change

Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism $\kappa$...
8
votes
1answer
610 views

Is there a notion of a “perfectly regular” topological space?

The separation axioms have exploded a little since the original list of four! Amongst them can be found "completely regular" spaces and "perfectly normal" spaces. The former is well-known: a point ...
3
votes
0answers
118 views

More on continuous images of dense orders

In this question I asked if there was an analogue of connectedness which applied to dense orders which were not required to be complete. Between them, Noah and Joel showed that every (infinite) ...
5
votes
1answer
227 views

When can boundedness be characterized topologically in Metric spaces?

Let H be a separable and infinite-dimensional Hilbert space. Is every closed subset of H homeomorphic to some closed and bounded subset of H?
3
votes
2answers
323 views

What, if anything, can be said about continuous images of densely ordered spaces?

If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove ...
5
votes
1answer
261 views

Solenoid of a continuous map of a ball, is it contractible?

Let $B$ be the closed unit ball in $\mathbb R^n$ and $f\colon B\to B$ a continuous map. Consider the infinite product $B^{\mathbb Z}$ equipped with the product topology. Consider the solenoid $$ S_f=\...
0
votes
2answers
470 views

Partition into connected sets by proximity

Let (δ;U) is a proximity space. I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y. I will call connected component a maximal connected set. Is this true: U is ...
0
votes
1answer
133 views

Connectedness of a union regading a proximity

Let δ is a proximity. I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y. Question: Let A and B are sets with non-empty intersection. Let both A and B ...
4
votes
2answers
413 views

A family of subsets with a “gluing” property

Somewhat in line with this previous MathOverflow question: I'm looking at a combinatorial structure consisting of a finite set $S$ of objects, and a family $F$ of designated subsets of $S$. We call ...
1
vote
4answers
5k views

Does Cauchy continuity imply uniform continuity? [No.] [closed]

It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff $$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...
6
votes
14answers
3k views

Applications of compactness [closed]

Similar to this question: Applications of connectedness I want to collect applications of compactness. E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...
4
votes
2answers
320 views

profinite spaces are the pro-completion of finite sets

The title sounds tautological, right? Perhaps I'm missing something completely trivial here ... Assume $X$ is a compact totally disconnected hausdorff space. It is known that $X$ can be written as ...
11
votes
3answers
4k views

Defining Quotient Bundles

This is an extremely elementary question but I just can't seem to get things to work out. What I am looking for is a natural definition of the quotient bundle of a subbundle $E'\subset E$ of $\...
10
votes
1answer
853 views

Category Theory / Topology Question

Let me begin by noting that I know quite little about category theory. So forgive me if the title is too vague, if the question is trivial, and if the question is written poorly. Let $\mathcal{C}$ ...
17
votes
3answers
2k views

How many tacks fit in the plane?

Call a tack the one point union of three open intervals. Can you fit an uncountable number of them on the plane? Or is only a countable number?
4
votes
2answers
457 views

How do you know when a reflective subcategory of Top is quotient-reflective?

A subcategory $\mathcal{C}$ of the category $Top$ of topological spaces is a reflective subcategory if the inclusion functor $i:\mathcal{C}\hookrightarrow Top$ has a left adjoint $R:Top\rightarrow \...
3
votes
2answers
856 views

Finite Topology vs sigma Field

Suppose we have a finite $\sigma$ -field $S$, of which $A$ and $B$ are member sets. Since $S$ is closed under union and complementation [by definition], it follows that $(A' \cup B')' = (A \cap B)' \...
44
votes
5answers
6k views

Fundamental group as topological group

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 ...
20
votes
15answers
3k views

Applications of connectedness

In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line). What are nice examples of applications of the idea of ...
20
votes
1answer
718 views

Is there a category of topological spaces such that open surjections admit local sections?

The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of $...
0
votes
2answers
386 views

Is there good evidence that topological spaces are the correct way to study the general theory of continuity? [closed]

My reason for asking is that the theory of metric spaces is so clean and so many significant theorems can be proved for an arbitary metric space (which makes it plausible to me that metric spaces are ...
5
votes
4answers
3k views

Connectedness and the real line

It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be ...
0
votes
1answer
644 views

Besicovitch Covering Constant for R^1

In the case where $E\subset\mathbb{R}^1$, a Besicovitch cover of $E$ is a cover by open intervals such that each point of $E$ is the center of some interval in the cover. The Besicovitch Covering ...
30
votes
14answers
11k views

What is your favorite proof of Tychonoff's Theorem?

Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by Loomis: http://www.archive.org/details/introductiontoab031610mbp http://ia331316.us.archive.org/3/...
32
votes
5answers
3k views

When factors may be cancelled in homeomorphic products?

It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, $A=B^{...
22
votes
8answers
3k views

Connections between ultrafilters in topology and logic

I have a some-what vague question. It seems to me that there are two main ways in which ultrafilters (on a set) can be used. One is in topology. The notion of an ultrafilter converging to a point is ...
1
vote
1answer
442 views

An example of a space which is locally relatively contractible but not contractible?

A space $X$ is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the ...
18
votes
4answers
3k views

Why are topological ideas so important in arithmetic?

For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in ...
39
votes
4answers
4k views

Are the rationals homeomorphic to any power of the rationals?

I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (...
18
votes
3answers
5k views

sets with positive Lebesgue measure boundary

Consider a compact subset $K$ of $R^n$ which is the closure of its interior. Does its boundary $\partial K$ have zero Lebesgue measure ? I guess it's wrong, because the topological assumption is ...
11
votes
6answers
18k views

How to understand the concept of compact space [closed]

the definition of compact space is: A subset K of a metric space X is said to be compact if every open cover of K contains finite subcovers. What is the meaning of defining a space is "compact". I ...
2
votes
2answers
2k views

Maximum number of shortest-paths

I would like to know if there is a equation for the maximum number of shortest paths that pass through r where r is a node contained in any path from node s (a fixed node, i mean, s is the only source ...
3
votes
1answer
744 views

Do continuous maps give continuity in the 'topology' of Hausdorff distance?

I was reading this question: limiting behaviour of converging loops on a torus And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their ...
4
votes
2answers
624 views

Determining if two algebraic sets are homeomorphic

Is there an algorithm which, given two polynomials in $n$ variables with real coefficients, $p(x)$, and $q(x)$, will determine whether the zero sets $p^{-1}(0), q^{-1}(0)\subset R^n$, are homeomorphic ...
17
votes
3answers
1k views

How thinly connected can a closed subset of Hilbert space be?

Let H be a separable (and infinite-dimensional) Hilbert space. Is it known whether there exists an infinite subset C of H with the following properties.? (1) C is connected and closed in H. (2) No ...
7
votes
1answer
649 views

Counting submanifolds of the plane

After thinking about this question and reading this one I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane. My ...
4
votes
1answer
2k views

Algorithms for the Lakes of Wada

The Lakes of Wada partitions the unit square in to three regions, all of whom share a common boundary. The Wikipedia entry (http://en.wikipedia.org/wiki/Lakes_of_Wada) gives a construction approach, ...
24
votes
3answers
4k views

Is “compact implies sequentially compact” consistent with ZF?

Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
6
votes
3answers
1k views

Products of Baire spaces

I could not find any references about this fact. I apologize if this is completely trivial, but is the product of two Baire spaces, or for that matter of finitely many of them a Baire space? Now is a ...
5
votes
2answers
804 views

Quantitative questions about the size of a finite epsilon net

Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...