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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2
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3answers
41 views

(Non-Euclidean) Compactness

Compactness in Euclidean Space The only definition of compact set that ever made sense to me was the intro calculus one: A set is called compact if it is closed and bounded. ...
-5
votes
0answers
29 views

The intersection of dense subset and open subset [on hold]

Let $A$ be a dense subset of $X$, and $B$ let be a non-empty open subset of $X$. Prove that $A\cap B \not = \emptyset $. Thank you very much
0
votes
0answers
14 views

Relationships Between Moduli Space and Objects They Parametrize

My friend and I were wondering recently what, if any, are the relationships between the geometric properties of a moduli space and the geometry of the objects that the space parametrizes. As an ...
1
vote
1answer
38 views

Question on one point compactification

I was given the following question in my general topology class assignment which is multi parts - most of which I managed alright by myself some of which I need help on. We are given a non compact ...
1
vote
1answer
91 views

Can something contain iteself? [on hold]

I asked this over on the Phyisics part of StackExchange, and they suggested I move my question here. And said question is: Can something contain itself? The question is simple enough, and I can ...
3
votes
4answers
87 views

Showing $\lbrace (x,y) \in \mathbb{R}^2:xy=1 \rbrace$ is Closed

Let $K=\lbrace (x,y) \in \mathbb{R}^2:xy=1 \rbrace \subseteq \mathbb{R}^2$. Show that $K$ is closed. I am following Munkres' topology book, and this is a step towards finishing problem 3 on p. ...
2
votes
1answer
20 views

prove finite intersection property for compact sets using sequential compactness

Prove finite intersection property for compact sets in metric spaces using sequential compactness with a direct proof . One approach is to prove sequential compactness and covering compactness are ...
0
votes
0answers
22 views

converging subsequences of two metrics

if $d$ and $d'$ are two metrics on a space $X$, is it true that they induce the same topology if and only if they have the same converging sequences ?
-1
votes
1answer
38 views

Why is there a subsequence of $(x_n)$ that converges to some point $y$ in $\mathbb R^p$?

A subset $A\subseteq\mathbb R^p$ is compact iff for every sequence $(x_n)$ in $A$ there is a subsequence $(x_{n_k})$ which converges to a point of $A$. I understand the whole proof of the above ...
0
votes
0answers
14 views

Definition of normal sets and compactness

I am struggling a little bit with this notion. In Conway's Functions of One Complex Variable, he offers the definition: A set $\mathscr F \subset C(G,\Omega)$ is "normal" if each sequence in ...
0
votes
3answers
55 views

Open sets and compact spaces

I am reading through Rudin's Principles of Mathematical Analysis and had a few related questions. First, Rudin defines an open set, $E$, as a set such that every point is an interior point. A point ...
1
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0answers
26 views

A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
1
vote
1answer
19 views

Continuity of multivariable functions

I have a question regarding norms on $\Bbb R^{n}$ and proving the continuity of multivariable functions. Specifically, suppose we have $f: \Bbb R^{2} \to \Bbb R$, for example. To prove $f$ is ...
1
vote
2answers
48 views

Show that $A$ is open in $\mathbb R$

I got this question in a test earlier today. I know it is a very small question, since it only counted 2 marks, but for some reason I simply could not get it?? Let $f:\mathbb R \to \mathbb R$ be ...
5
votes
1answer
39 views

Compact subsets of the space of real functions $\mathbb{R}^\mathbb{R}$

I was suprised that this question hasn't been asked - or maybe it was, but asked differently. Anyway, I want to characterize the compact sets in the space of real functions $\mathbb{R}^\mathbb{R}$ ...
4
votes
2answers
194 views

Space which is neither locally connected at any point nor totally disconnected

Let $X$ be a topological space; then we say that $X$ is locally connected at $x$ if $x$ admits a neighborhood basis of open, connected sets. In this sense, a space is locally connected iff it is ...
2
votes
1answer
34 views

Show that if $f$ is a proper,surjective map which is locally injective then $f$ must be a covering map

Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map. It is well ...
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votes
0answers
31 views

Finite subset of $\Bbb R$ is nowhere dense [on hold]

I need to show that every finite subset of $\Bbb R $ is nowhere dense. Thanks
0
votes
0answers
22 views

Example of the inequality $c_0\neq\bigcup l_p$

As part of an exercise, I was asked to prove or disprove the following proposition: There exists an $x\in c_o$, such that $x\notin l_p$ for every $1\le p\lt\infty$. Before I show my proof, I will ...
7
votes
1answer
45 views

Irreducible projective cubic, exists continuous surjection?

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
3
votes
1answer
27 views

Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, ...
1
vote
4answers
53 views

What is induced topology?

In my text, it says "Given a topological space $X$ and a subspace $S ⊂ X$, define the induced topology on $S$ to be the topology in which the open sets are of form $U ∩ S$, where $U$ is open ...
1
vote
1answer
22 views

Product (arbitrary) of open functions is open.

Let $f_{\alpha}\colon X_{\alpha}\to Y_{\alpha}$ be open, for all $\alpha \in J$. Then $\prod_{\alpha} f_{\alpha}\colon \prod_{\alpha}X_{\alpha} \to \prod_{\alpha}Y_{\alpha}$ is open? Both $ ...
1
vote
1answer
37 views

Box topology and axiom of choice

Below is the definition of box topology: Given an indexed family of topological spaces $X_\alpha $, the collection of all sets of the form $$\prod_{\alpha\in J} U_\alpha,$$ where $U_\alpha$ is open ...
0
votes
0answers
21 views

It is correct this definition of limit of a function?

I have a definition of the limit of a function in some point $\alpha$ for metric spaces on this manner: We have two metric spaces $(E,d)$ and $(F,p)$; $A\subset E$, $f:A\to F$. Then ...
5
votes
1answer
61 views

Unifying Connection Between Topological Embeddings and Quotient Maps

In a book on topology I'm reading the following theorem seemed striking to me, not for its proof, which I believe I understand, but because there's some nice symmetry going on that I'd perhaps like ...
1
vote
1answer
37 views

An exhaustive continuous map is a covering map.

$p_1:\tilde X_1 \rightarrow X \, ; \, p_2:\tilde X_2 \rightarrow X$ two coverings maps, where $X$ connected and locally path-connected, and suppose that $f:\tilde X_1 \rightarrow \tilde X_2$ is an ...
3
votes
1answer
54 views

Necessarily a homeomorphism?

Let $D$ be the projective curve defined by $y^2z = x^3.$ Consider the map $f: \mathbb{P}_1 \to D$ defined by$$f[s, t] = [s^2t, s^3, t^3].$$Is it necessarily a homeomorphism? Any help would be greatly ...
0
votes
1answer
19 views

General Form of a Open Set in the Product Topology in a Countably Infinite Product.

Suppose $\{X_n\}_{n\in\Bbb N^+}$ is a family of topological spaces. I understand that a typical basis element of the product topology has the form $$\prod_{n=1}^k U_n\times\prod_{n=k+1}^\infty ...
0
votes
0answers
42 views

The degree-genus formula cannot be applied to singular curves in $\mathbb{P}_2$?

(The degree-genus formula) The Euler number $\chi$ and genus $g$ of a nonsingular projective curve of degree $d$ in $\mathbb{P}_2$ are given by$$\chi = d(3-d)$$and$$g = {1\over2}(d-1)(d-2).$$ My ...
1
vote
0answers
23 views

Which definition of a neighborhood is more standard? [duplicate]

I came across the following two definitions of a neighborhood in a topological space $X$. Definition: A set $N\subset X$ is a neighborhood of $x\in X$ if $N$ contains a open set in $X$ which ...
1
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1answer
36 views

Subspaces that undo Products

I have been working on Munkre's homework sets, and I have come across the following phenomenon: Let $\mathbb{R}_\ell$ be the lower limit topology on the real numbers. If you consider a line as a ...
4
votes
0answers
41 views

Any two maps to a cone space are homotopic.

I have to prove that any two continuous functions to a cone space are homotopic. Definition of cone space: If $Y$ is any topological space and $I=[0,1]$ is the closed unit interval in $\mathbb R$, ...
4
votes
1answer
51 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
2
votes
2answers
53 views

constructing a CW Complex

I am looking at an example of constructing a CW complex for a space X. The example i am looking at is that for The quotient of $S^2$ obtained by identifying north and south poles. The solution is as ...
0
votes
1answer
29 views

metric space: equivalence of several mertric.

I have two questions: Q1) Are all metric on a metric space are equivalent ? Q2) If not: Let $d_1,d_2$ two metric on $X$. If something has a property with a $d_1$ will it hold for $d_2$ too ? For ...
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vote
2answers
42 views

Finding a choice for Epsilon for open/closed set proofs

I'm studying the proofs for open/close sets by using the following definition: I'm having problems to understand the proofs. The proofs sounds pretty straightforward: just choose a value for ...
-2
votes
1answer
16 views

an equivalent condition for compactness of a metric(topologic) space.

Let $X$ be a metric(or topological) space. 1) If every continuous function $f:X \rightarrow \Bbb R$ has a bounded image, then is $X$ a compact space? 2) If every continuous function $f:X \rightarrow ...
0
votes
2answers
21 views

A question involving continuity with respect to the product topology

Let H be a nonempty set, $\cdot$ a binary operation on H, $\Gamma$ a topology on H and $$\varphi : H \times H \to H, \;\; \varphi(x, y) = x y, \;\; \forall x, y \in H$$ continuous with respect to the ...
-1
votes
0answers
23 views

Idempotent ideal in ring of continuous functions

Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be an idempotent ideal?
2
votes
2answers
57 views

Homeomorphism on the Hilbert space

We can consider two different topologies on the Hilbert space ; $l^{2}(\mathbb{N})$. One is the topology deduced from the norm \begin{equation*} \|f\|=\sqrt{\sum_{n=1}^{\infty} f(n)^{2}}, ...
6
votes
1answer
60 views

Does every continuous map between compact metrizable spaces lift to the Cantor set?

I'm interested in the universal properties of the Cantor set. It is well-know that the Cantor set $2^\mathbb{N}$ is "universal" in the category of metrizable compact spaces, in the sense that every ...
2
votes
2answers
130 views

$\epsilon$-dense subsets on $\mathbb R/\mathbb Z$.

Let $\langle M, d\rangle$ be a metric space. We say that $A \subset M$ is $\epsilon$-dense if every open ball of radius $\epsilon$ contains a point of $A$. Now let $T=\mathbb R/\mathbb Z$, the ...
0
votes
3answers
24 views

A countable Tychonoff space is normal?

I am trying to prove a countable tychonoff space must be normal but I cannot. Here is my work so far: We take two disjoint closed sets $F_1$ and $F_2$. Since $F_1$ is countable and $F_2$ is closed ...
3
votes
1answer
39 views

Explicit homeomorphism between open and closed rational intervals?

Sierpiński's theorem states that every countable metric space without isolated points is homeomorphic to $\mathbb{Q}$. (A proof can be found here and a discussion here). An immediate corollary is ...
2
votes
0answers
51 views

What's the most general geometry branch?

What is the most general geometry of curves and surfaces? For example, at curves, we define in differential geometry the tangent vector as the derivative of a regular curve, but visually many other ...
0
votes
1answer
33 views

$\mathbb{R}^{2}$ and $\mathbb{R} \times [0, +\infty]$ are homotopy equivalent, but not homeomorphic

So, let's consider $M=\mathbb{R}^{2}$ and $N= \mathbb{R} \times [0, +\infty]$ - two topological spaces. Since $\pi_{1}(M)=\pi_{1}(\mathbb{R}) \times \pi_{1} (\mathbb{R}) = \{0 \}$ (since $\mathbb{R}$ ...
0
votes
0answers
24 views

Does total order imply linearisation

Suppose $X$ is a totally ordered set. Does this mean that $X$ can always be linearised? I mean can $X$ be always written in a linear order like $\mathbb{R}$ ? I came across this question when I was ...
0
votes
2answers
38 views

Group homomorphism on unit circle

For $n\in \mathbb{Z}$, define the map $f_n:S^1\to S^1$ as $f_n(z)= z^n$, where the unit circle $S^1$ is observed as the subspace $\{z\in\mathbb{C}|\ |z|=1\}$. How would one compute the induced group ...
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vote
2answers
24 views

Extreme value theorem, without Heine Borel.

I was wondering, if there are any mistakes, in this proof of the extreme value theorem: Theorem. Let $X$ be a compact set and $f:X\rightarrow\mathbb{R}$, s.t. $f$ is continuous. Then there exists ...