Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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6
votes
1answer
121 views

Difference between X/A and G/H

I am primarily a student of physics and am trying to self-learn some algebraic topology. I am having some difficulty understanding the differences between the constructions of $(X,A)$ (Pair of ...
10
votes
3answers
766 views

homeomorphism of topological spaces is an equivalence relation ?

Would it be ok to say that homeomorphism of topological spaces is an equivalence relation ? I know that there isn't a base "set of all topological space" but since I encountered this phrase in ...
1
vote
0answers
69 views

Projection maps of products of functors

Let I be a be a partially ordered set such that for any $i$; $i'$ $\in$ I, there exists $i''$ $\in$ $I$ such that $i'' > i, i'$. Let F be a functor from $I^{op}$ to finite ...
0
votes
1answer
218 views

Definition of Genus

Are genus necessarily toral -- as shown in the illustration on wiki what about a tube, does it qualify for having genus 1? What about this? Does this have genus 1 or 2? Thanks.
1
vote
3answers
495 views

Lack of homeomorphism between compact space and non-Hausdorff space

Show that a continuous bijection $f : X \to Y$ with $X$ compact and $Y$ Hausdorff is a homeomorphism. Give an example to show that such a continuous bijection is not necessarily a homeomorphism if $Y$ ...
28
votes
2answers
580 views

Existence of non-constant continuous functions

Under what circumstances is there at least one non-constant continuous function from a topological space $X$ to a topological space $Y$? Assume that $X$ and $Y$ each have at least two points. If $X$ ...
0
votes
1answer
75 views

Is the space of continuous maps $Top(X,Y)$ between two topological spaces compact if $X$ is?

Suppose that $X$ and $Y$ are topological spaces, and we consider $Y^X$, i.e. the space of maps from $X$ to $Y$ with the compact-open topology. If $X$ is compact then can we say anything about $Y^X$? ...
1
vote
2answers
599 views

Compact but not sequentially compact question

At this page: http://planetmath.org/encyclopedia/SequentiallyCompact.html you can find an example of a compact but not sequentially compact space. My question is: how to prove the existence of "$r ...
1
vote
1answer
217 views

Does perfectly normal $\implies$ normal?

A question here on perfectly normal spaces got me into investigating the definition of such a space. The definition on wikipedia says A perfectly normal space is a topological space X in which ...
2
votes
0answers
319 views

Do we need net refinements not induced by preorder morphisms?

From Engelking's book on general topology (slightly rephrased): Definition: We say that the net $S': \Sigma' \to X$ is finer than the net $S: \Sigma \to X$ if 1. there exists a function $f: ...
3
votes
2answers
303 views

Minimal dense subset of $\mathbb{Q} \cap [0,1]$

The following question was a problem in an Analysis exam: Let $n \in \mathbb{N}$. Define $A_{n} := \displaystyle \left\{\frac{k}{2^n} \bigg| k \in \mathbb{Z}, 0 \leq k \leq 2^n \right\}$. Let ...
9
votes
1answer
1k views

Product and Box Topologies

I am having a hard time understanding why the box topology is finer the the product topology. Of course I know that with finite product, the two are the same. With product topology, the basis elements ...
7
votes
2answers
356 views

Is there a countable, regular space with no isolated points which is not homeomorphic to the rationals?

It's known that every countable, metrizeable space with no isolated points is homeomorphic to the rationals with the standard topology. Suppose you wanted to reformulate the above without referencing ...
6
votes
1answer
407 views

Why are these two definitions of a perfectly normal space equivalent?

I've been skimming through some topology textbooks recently. Some sources, (such as Munkres' Topology and Willard's General Topology) define a space $(X,\mathcal{T})$ to be perfectly normal iff $X$ is ...
4
votes
4answers
288 views

Is there an isomorphism from $[0,1]$ to $[0,1]^2$

Is there any such isomorphism? Or between any closed interval on $\mathbb{R}$ and any closed simply connected subset of $\mathbb{R}^2$? If so, how is it expressed and does it preserve any structure? ...
8
votes
1answer
742 views

If derivative of a function is the zero function in $\mathbb R^n$, then the function is constant when the domain is path-connected

Some definitions first. Let $A \subseteq \mathbb R^n$. Let $x,y \in A$. A path between $x$ and $y$ is a continuous function $f: [0,1] \rightarrow \mathbb{R}^n$ with $f(0) = x$ and $f(1) = y$. The set ...
3
votes
1answer
222 views

Axiom of choice and Topology?

I was just thinking if you need the axiom of choice to see if some topologies are actual topology. $\cup_{\lambda \in \Lambda} U_{\lambda} \in \tau$ See if you have $\tau$ is uncountable and ...
4
votes
1answer
338 views

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?
1
vote
1answer
116 views

Why are such functions not always necessarily uniformly continuous?

One form of Urysohn's lemma (I suspect there may be more than one) is that on a normal space $(A,\mathcal{T})$ with disjoint closed sets $X$ and $Y$, there exists a continuous real valued function $f$ ...
4
votes
1answer
156 views

Are there disjoint closed sets which contain sequences that get arbitrarily close to each other?

I had a question of curiosity. Take the interval $(0,1)$ with the usual metric in $\mathbb{R}$. Is it possible to find closed sets $X$ and $Y$ with $X\cap Y=\varnothing$ such that there is a sequence ...
1
vote
2answers
191 views

Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the standard topology on $\mathbb R$?

Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the topology generated by open intervals of $\mathbb R$ on $\mathbb R$? I'd just like to know if ...
2
votes
0answers
135 views

Compact Metric Spaces and Properties? [duplicate]

Possible Duplicate: Condition for family of continuous maps to be compact? I was reading through general-topology posts and I couldn't quite get this one. Here's a reformulation of the ...
4
votes
1answer
674 views

The uniform metric on $\mathbb{R}^\omega$

I'm working on a question from Topology by Munkres on p. 127 (Exercise #6). Here it is: Let $\bar{\rho}$ be the uniform metric on $\mathbb{R}^\omega$. Given $\mathbb{x}= (x_1, x_2, ...) \in ...
7
votes
1answer
708 views

Characterisation of compact subsets of Banach spaces

I have the following homework question: Characterize the compact subsets of the following Banach spaces: (1) The space $c_0$ of null sequences (that is, sequences $(x_n)$ of scalars with $ | ...
5
votes
1answer
105 views

Metrisability of an Arbitrary Topological space

Is there a general condition to tell whether a topological space is metrisable? (Without having to find the metric explicitly?) Thanks
5
votes
1answer
274 views

Zariski topology over $\mathbb R$

What is a "Zariski topology on $\mathbb R$"? I don't think I quite understand the definition of a "Zariski topology". Thank you.
5
votes
1answer
2k views

Proving that this set is compact using the open cover definition

Let $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ be a subset of $\mathbb{R}$. I have to prove using the open cover definition that this is compact. Could you help me, please?
0
votes
1answer
154 views

Convergence of the sequence $f_n : [0,1] \rightarrow \mathbb{R}$ defined by $f_n(x)=x^n$

From introductory real analysis, I know that the sequence of functions $f_n(x)=x^n$ converges pointwise in that $f_n \rightarrow 0$ for $0 \leq x < 1$ and $f_n \rightarrow 1$ whenever $x=1$. Thus, ...
5
votes
1answer
177 views

Quotient of zero-dimensional hausdorff space

I've read (in joy of cats) that every topological space is a regular quotient of a zero-dimensional hausdorff space. So far, I could not find a proof. Do you know one, or a reference?
0
votes
1answer
143 views

Can every closed curve be modified in the following way to produce a simple closed curve?

Is there a sequence of the following operation that change a closed curve with finite number of self-intersections to a simple closed curve? Also, every self-intersection differs at least $\epsilon$ ...
2
votes
2answers
165 views

Interior points of correspondence

Let $\Gamma(x)$ be a correspondence (i.e. a set-valued function) between two Euclidean spaces which is continuous (i.e. both lower- and upper-hemicontinuous). If $y$ is a point in the interior of ...
3
votes
3answers
189 views

Connected components of subspaces vs. space

If $Y$ is a subspace of $X$, and $C$ is a connected component of $Y$, then C need not be a connected component of $X$ (take for instance two disjoint open discs in $\mathbb{R}^2$). But I read that, ...
2
votes
1answer
329 views

Isolated points of compact subset of the rationals

Does there exist a compact, nonempty subset of the rationals without isolated points ? My motivation was the following: If so one could define a map $f$ from the set of all compact subsets of the ...
4
votes
1answer
293 views

Can anyone tell me why the arclength integral is a lower semicontinuous function on the set of continuously differentiable real-valued functions?

I posted the question stating that it was upper semicontinuous, but that was definitely wrong. I am trying to prove lower semicontinuity.
15
votes
2answers
920 views

Origins of the modern definition of topology

The modern definition of topology is 'a family of subsets of a set $X$ containing the empty set and $X$, closed under unions and finite intersections'. In Grundzüge der Mengenlehre (1914) Hausdorff ...
11
votes
4answers
916 views

Showing that $\mathbb{R}$ is connected

So I know that $\mathbb{R}$ is both open and closed. But given a set, $X\subset \mathbb{R}$, $X\ne \emptyset $ that is both open and closed, how does one show that $X=\mathbb{R}$? Here is my ...
5
votes
1answer
206 views

Looking for non-trivial topologies satisfying certain conditions

I'm looking for topologies T on an infinite space X which divide the subsets of X into 2 non-empty collections: (1) sets which are both open and closed (clopen); (2) sets which are neither open nor ...
3
votes
4answers
428 views

what exactly is an open set?

Many, infact all the books on topology I have come across define open sets in the following way: "A set $A$ is said to be open if by moving in small amounts in any direction about any point we ...
0
votes
2answers
162 views

Can anyone tell me why the arclength integral is an uppersemicontinuous function on the set of continuously differentiable real-valued functions?

The continuously differentiable functions are equipped with the topology induced by the sup norm. I know that I can make the arclength integral close to the arclength of a piecewise linear function. ...
1
vote
1answer
230 views

Construction of a noncommon continuous real function

First a simple definition. We say that a point $x$ is a local strict maximun (LSM) of $f$, if there exists a $\delta > 0$ such that $$ 0 < |x-y| < \delta \implies f(y) \lt f(x). $$ One can ...
1
vote
1answer
261 views

Condition for family of continuous maps to be compact?

A standard result in topology is that if $(K,\mathcal{T})$ is a compact space, and $f$ a continuous surjective map into another space $L$, then $L$ is compact. I'm curious about what happens when we ...
6
votes
3answers
402 views

Range of curve on a compact interval is nowhere dense

I glanced through this question on why $\mathbf{R}^2$ is not of the first category. I understand how this would follow if the image of a curve on a compact/finite interval in $\mathbf{R}$ is nowhere ...
4
votes
1answer
104 views

Is the pre-image of a cellular map a CW complex?

In general, if we have a map between CW-complexes, $f:X\to Y$, and $f$ is cellular, then is is clear that $f^{-1}(Y)$ (the inverse image, $f$ is not invertible in general) is also a CW-complex? ...
3
votes
1answer
491 views

Prove that the convex hull of a set is the smallest convex set containing that set

How do you prove that the convex hull of A is the smallest convex set containing A? edit: definition of a convex hull: Given a set A ⊆ ℝn the set of all convex combinations of points from A is called ...
14
votes
5answers
778 views

Why isn't $\mathbb{R}^2$ a countable union of ranges of curves?

I came across this question on the topology board at AoPS, where it hadn't really received an answer. It seems interesting, but I'm not sure how to solve it. Hopefully an answer will be found here. ...
3
votes
1answer
112 views

Exact preimage of an interesting open ball

Consider the function $$g: \mathbb{R}\to \mathbb{R}^\omega$$ given by $$g(t)=(t,t,t,...)$$ where $ \mathbb{R}^\omega$ is in the uniform topology. Can we find the exact answer to ...
3
votes
5answers
1k views

Complete implies locally compact in length metric space?

I am confused. The way I see it, in a complete metric space, closed balls of finite diameter are compact since they are complete and totally bounded. Consequently a complete metric space is locally ...
2
votes
3answers
121 views

The continuity of a function in the uniform topology

I am doing the exercises in the book Topology(2nd edition) by Munkres. Here is my question(page 127, question 4(a)): Let $h:R\to R^\omega$ be a function defined by $h(t)=(t, t/2, t/3, \ldots)$ where ...
2
votes
1answer
56 views

Follow-up: that the embedding is the intersection of closed sets

I was reading through general-topology posts and I couldn't quite understand this one. I tried asking directly on the thread, but I didn't get a response. This is in reference to Professor Israel's ...
1
vote
0answers
170 views

homework problem about the projective real space

Sorry for ask this problem, but I am very complicated with this problem :/ . My course it´s of topology, the teacher said that we only need the definition of the quotient topology and of $$ P_R^2 ...