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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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 ...
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3answers
41 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 ...
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20 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?
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1answer
15 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 ...
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2answers
41 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 ...
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1answer
21 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 $\mathbb{R}^\mathbb{R}$ ...
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2answers
186 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 ...
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1answer
30 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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26 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
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21 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 ...
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1answer
32 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 ...
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1answer
25 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 $\partial D_1\cup\{\infty\}$, ...
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50 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 ...
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1answer
20 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 $ ...
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1answer
34 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 ...
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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 ...
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1answer
53 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 ...
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1answer
29 views

A question about a homeomorphsim

I have a question about the following example from Munkres' book "Topology". Is the part that I underlined "vice versa" necessary to conclude that $F$ is a homeomorphism of $(-1,1)$ with $R$? It ...
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1answer
29 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 ...
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1answer
50 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 ...
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1answer
18 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 ...
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36 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 ...
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22 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 ...
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1answer
34 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 ...
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40 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 $\mathbb R$, ...
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34 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 ...
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2answers
50 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 ...
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1answer
28 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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2answers
40 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 ...
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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 ...
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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 ...
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21 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?
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2answers
56 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}}, ...
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1answer
56 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 ...
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2answers
129 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 ...
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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 ...
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1answer
38 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 ...
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49 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 ...
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1answer
32 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 $\mathbb{R}$ ...
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0answers
23 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 ...
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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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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 ...
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1answer
51 views

The union of a sequence of closed sets with empty interiors has empty interior in a compact Hausdorff space?

This is problem 5 in section 27 of Munkres' TOPOLOGY, 2nd ed Let $X$ be a compact Hausdorff space; let $\{A_n\}_{n\in \mathbb{N}}$ be a countable collection of closed sets of $X$. If each set $A_n$ ...
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1answer
48 views

Does proper map $f$ take discrete sets to discrete sets?

Suppose $f:X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Are the following results true? $1$. The map $f$ takes discrete sets to discrete sets. $2$. If $f$ is ...
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1answer
47 views

Prove that the convergent sum of a real sequence is a metric

I want to show that $$ \varrho(\{a_n\},\{b_n\})=\left(\sum_{n=0}^\infty{(a_n-b_n)^2}\right)^{1/2} $$ is a metric, where $\{a_n\}_{n\in\Bbb N}\in \ell_2$, and $\ell_2$ is the set of all real sequences ...
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1answer
55 views

Questions about a topological proof of the FTA

I'm a high school student, curious about proofs of the Fundamental Theorem of Algebra. Specifically, I've been thinking about one of the topological proofs of the theorem, given in Courant's book, ...
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1answer
31 views

Non-empty intersection of specific sets

For any set Y (to begin with, it may be countable), given a collection of relations $$R = \{R_y \subseteq \{0,1\}^Y \mid y \in Y\},$$ having the finite intersection property and such that for ...
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1answer
25 views

Every nontrivial linear functional is open

Let $X$ be a normed linear space and let $f:X\to \mathbb K$ be a nontrivial linear functional. I want to prove that $f$ is open. I tried as follows: Let $E$ be an open set in $X$ and let $y\in f(E)$. ...
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1answer
43 views

does the closure of interior of a set equal to closure of this set?

Does the $\text{Cl}(\text{Int} A)=\text{Cl}(A)$? Here "Cl" denotes closure, "Int" denotes interior. That is not a duplicate of the question of "does the closure of interior of a set equal t the ...
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3answers
54 views

Does there exist a surjective continuous map $D^2 \to S^1$?

By considering the induced homomorphism on the fundamental groups, we know that there is no retract $D^2 \to S^1$. But is there any continuous surjection from $D^2$ to its boundary? It seems unlikely ...