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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1answer
25 views

Is the surface of a torus 2-dimensional?

Unless I'm very mistaken, the surface of a torus is 2-dimensional, as is the surface of a sphere. The reason being that being on the surface you can only move in 2 dimensions, up or down is not well ...
7
votes
5answers
969 views

Is a ball always connected in a connected metric space?

If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?
2
votes
1answer
32 views

Graph of a $G_\delta$-function

Let $f:\mathbb R \to \mathbb R$ be a function. It is well known that if $f$ is continuous ($f^{-1} [A]$ is closed whenever $A$ is closed), then its graph is closed in $\mathbb R ^2$. Here is an ...
0
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2answers
63 views

Does there exist a Vector that can't be written as a Tuple of Scalars?

The most abstract/general definition of a vector The most general definition of a vector is as an element of a vector space. Given a vector $u$, we can always say that there exists a vector space $V$ ...
0
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1answer
24 views

Determine If This Is A Topology on U

Let $f:U \to V$ be a function and supposed that $T$ is a topology on $V$. Then {$f^{-1}(S):S \in T$} is a topology on $U$. I understand that I need to prove this or provide a counterexample using ...
4
votes
1answer
62 views

How many connected components are left after removing a line from the plane?

Let $A \subseteq \mathbb{R}^2$ be a subset of the plane which is homeomorphic to $\mathbb{R}$. How many connected components does $\mathbb{R}^2 \setminus A$ have? My conjecture is that only one or ...
2
votes
5answers
118 views

These two spaces are not homeomorphic…right?

why is $\Bbb R\times[0,1]\not \cong \Bbb R^2$? we can't use the popular argument of deleting a point and finding that one has more path components than the other here. So my idea is to delete a strip ...
2
votes
1answer
38 views

What is an example of a continuous but not closed function? [duplicate]

I have two questions about closed functions. Firstly, we say that a function is closed if it maps closed subsets in the domain to closed subsets in the co-domain. Polynomials are typical examples of ...
0
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2answers
46 views

If X is compact and $C(X)$ is the space of all continuous real valued functions. Prove $C(X)$ is a complete metric space.

Let $X$ be a compact metric space and define $C(X)$ to be the space of all continuous real valued functions on $X$ with a metric defined by $$d(f,g)=\sup_{x \in X} |f(x) -g(x)|.$$ Show that $C(X)$ is ...
-1
votes
1answer
48 views

Continuity/Inverse in Topological Spaces

Let $(X,d1)$and $(Y,d2)$ be topological spaces and suppose $f:X → Y$ is continuous on $X$ and $B ⊆ Y$. Prove or provide a counterexample for a and b. a. $f^{−1}(Int(B)) ⊆ Int(f^{−1} (B))$ b. $ ...
0
votes
2answers
24 views

Is giving an explicit homeomorphism sufficient to prove that there exists a homeomorphism?

I am asked to prove that a square is homeomorphic to a circle. Now we can construct the homeomorphism explicitly by first having a bijection $\gamma$ that takes an arbitrary square in $\mathbb{R}^2$ ...
0
votes
1answer
33 views

Reference request: alternative proof for every open set in $\mathbb{R}^n$ can be expressed as countable disjoint union of open boxes

A "box" is a cartesian product of intervals of the type $[a,b]$ I am using Terence Tao's introduction to measure theory and on page 24 a proof of title statement is given, however, it is quite ...
1
vote
2answers
24 views

Show that $\bigcup_{n=1}^\infty A_n= B_1 \backslash \bigcap_{n=1}^\infty B_n$

Let $\{B_n\}$ be a decreasing set $B_1 \supseteq B_2 \supseteq B_3 \supseteq ....$ Define $A_n = B_1 \backslash B_n$ i.e. $A_1 = \varnothing, A_2 = B_1 \backslash B_2$ If we imagine $\{B_n\}$ as a ...
1
vote
1answer
24 views

Continuous/Onto Function Between Topological Spaces

Let $(U,T)$ and $(V,D)$ be topological spaces, where $D$ is the discrete topology. Suppose that $f:U\to V$ be a function that is $T - D$ continuous. If $f$ is surjective and D is the discrete ...
1
vote
2answers
21 views

Continuity of a map in a metric space

Let $C^0([a,b])$ denote the space of continuous function $f:[a,b]→\Bbb R$. Define $ d(f,g)= \sup_{[a,b]}|f-g| $. We define $F:C^0([a,b])→\Bbb R$ to be $F(f)=\int_a^b f$. I want to show that $F$ is a ...
1
vote
2answers
41 views

Complete metric space

Let $C^0([a,b])$ denote the space of continuous function $f:[a,b]→\Bbb R$. Define $d(f,g)=sup_{[a,b]}|f-g|$. I've proved that d is metric in $C^0([a,b])$. How to prove that this metric space is ...
0
votes
1answer
37 views

If a subspace $(A, T_A)$ of a space $(X,T)$ is connected, then $A ∩ A' ≠ ∅$.

I start by assuming that $A \cap A' = \emptyset$, then if $x$ is in $A$, we have that $x$ is not in $A'$. Then there exists an open set $U$ of $T$ such that $A ∩ U = \{x\}$, but from there I can't ...
0
votes
2answers
19 views

Prove that the closure of a connected space is connected (Topology)

Let A be a connected subspace of a topological space (X,T), I start by assuming that cl(A) is disconnected. Therefore there exists two open sets of T (call them U and V) such that U ∩ V = ∅ and U ∪ V ...
1
vote
1answer
30 views

Let $S=\{( x,y) \in \mathbb{R}^{2}\mid xy>1\}$, show that $S$ is open

Let $S=\{(x,y) \in \mathbb{R}^{2}\mid xy>1\}$. Show that $S$ is open. Please step by step
2
votes
0answers
37 views

Intuition for universal quotient maps [migrated]

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman, the two characterizations of ...
0
votes
2answers
15 views

$\{g^{-1}(\Omega):\Omega\in\tau_Y\}\in \tau_X$ in topology for continuous functions

Let $g: X\rightarrow Y$ be continuous. Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be topological spaces. Why do we have $\tau=\{g^{-1}(\Omega):\Omega\in\tau_Y\}\subseteq\tau_X$? My attempt: Let ...
0
votes
0answers
10 views

Construct a specific base for Fine uniformities in the diagonal(Entourages) case

For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. To construct Fine uniformities, Let ...
1
vote
0answers
27 views

limit points in a topological space

I'm taking a topology course with the textbook by Munkres. And I saw the definition of convergence of a sequence and limit point of a subset of a given topological space. I have told in the class ...
5
votes
1answer
481 views

It there an example of sum and product of continuous functions is not continuous?

The sum and product of two continuous functions is continuous I can prove this easily when the space is metrizable, but I don't get it when the space is non-metrizable. Is there a counterexample ...
3
votes
0answers
29 views

An asymmetry in the Galois connection between topologies and sequential convergences?

On a set $X$ consider a relation $c \subseteq X^{\mathbb{N}} \times X$ and for $((x_n), x) \in c$ write $x_n \to_c$ x. Such a relation $c$ is a sequential convergence if (i) $x \to_c x$ for all $x \in ...
0
votes
1answer
57 views

How to show $\partial A = \varnothing \Rightarrow A=R^n$ [closed]

Let $A\subset R^n$ and dim$A=n$, $\partial A$ is the relative boundary of $A$. If $\partial A=\varnothing$ how to show $A$ is $R^n$ ? Picture below is from XX page of Schneider R.-Convex Bodies_ ...
1
vote
2answers
44 views

Prove that is $f:X\rightarrow Y$ is continuous and $\lim_{n\rightarrow\infty}x_n=x$ then $\lim_{n\rightarrow\infty}f(x_n)=f(x)$.

I am preparing for a midterm and came across this question from a past exam. I am hoping someone could help point me in the correct direction with the second part. Let $X$ and $Y$ be topological ...
0
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1answer
26 views

Basic Topology: Countable Complement Topology

Let $B$ be a subset of $\Bbb R$. In the topological space $(\Bbb R,\mathscr{T})$, if $B$ is not closed, then $B$ is dense. $\mathscr{T}$ is defined as the countable complement topology, which is ...
3
votes
3answers
218 views

Boundary is equal to its closure

Let $(X,T)$ be a topological space and let $S$ be a subset of $X$. Prove $Bd(S) = Cl(Bd(S))$. My initial thought is that the boundary is a closed set of points and the closure of a closed set is ...
2
votes
3answers
26 views

Proving continuity of the following function

Let $X,Y$ be compact sets in $\mathbb{R}^n$ (with the usual topology) and let $f:X\times Y \rightarrow \mathbb{R }$ be a continues function function moreover let $P(Y)$ be the space of all ...
0
votes
1answer
17 views

Cluster and limit points of $z$-ultrafilters

Let $X$ be a subspace of $Y$. Let $\mathcal{F}$ be a $z$-filter of $X$. We say that $y\in Y$ is a cluster point of $\mathcal{F}$ if $y\in\overline{Z}$ for every $Z\in \mathcal{F}$ (note that this ...
0
votes
1answer
41 views

Example of a disconnected manifold where the tangent space is not the dimension of the manifold?

Wikipedia says that the tangent spaces of a connected manifold all have the same dimension, equal to that of the manifold. Well, is there an example of a simple disconnected manifold that doesn't ...
-2
votes
2answers
35 views

Example of a space that is connected but not path connected? [duplicate]

Wikipedia says that path-connectedness is a stronger property than connectedness. My intuition cannot seem to come up with an example of an object that is connected but not path-connected. Are there ...
1
vote
1answer
45 views

A question about closed (but not necessarily compact) connected subsets of Euclidean spaces.

Is the following statement true?...... If $C$ is a non-degenerate closed and connected subset of the Euclidean plane $\mathbb R ^2$ and $p$ is any point of $C$, then there exists a connected ...
1
vote
1answer
35 views

Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.

I have some questions about this proof that "Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.": By the example (12), we just have to consider the ball $B(0,1)$, we ...
0
votes
2answers
58 views

Sequential compactness implies compactness: what is wrong with this argument?

Definitions A filter is a poset $(I,\leq)$ such that for any $\alpha,\beta\in I$ there is $\gamma\in I$ such that $\gamma\geq\beta,\gamma\geq\alpha$. A net in a set $X$ is a function from a poset to ...
1
vote
1answer
67 views

Powers-of-10-multiples of $\pi$ (or any irrational) are dense

Very related, but not the same, to this question Multiples of an irrational number forming a dense subset, is the next one: Is the sequence $(\{10^n\pi\})_{n=1}^\infty$ dense in the interval ...
1
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2answers
33 views

Deck transformation of the $n$-sheeted covering $\Bbb S^1 \to \Bbb S^1$

From Hatcher: For the $n$-sheeted covering space $S^1\to S^1$, $z\mapsto z^n$, the deck tranformations are the rotations of $S^1$ through angles that are multiples of $2\pi/n$. Why is this so? I ...
2
votes
2answers
28 views

Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
3
votes
2answers
45 views

When does convergence in quotient space $X / {\sim}$ induce convergence in $X$

Let $X$ be a topological space, $\sim$ an equivalence relation and $q : X \to X/{\sim}$ the quotient map that induces on $X/{\sim}$ the quotient topology. Assume that $p_n \to p$ in $X / {\sim}$. ...
0
votes
2answers
24 views

E is infinite subset of compact set, then is E' also a subset?

Here's a theorem in Rudin's Principles of Mathematical Analysis. 2.37 Theorem: If E is an infinite subset of a compact set K, then E has a limit point in K. Proof: If no point of K were a limit ...
1
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1answer
25 views

About Expansive homeomorphism

Let $f:X\rightarrow X$ be a homeomorphism. $f$ is called an $c$-expansive homeomorphism, whenever for every $x\neq y$, there is an integer $n$ with $d(f^{n}(x), f^{n}(y)) >c$. Question. Is there ...
1
vote
1answer
72 views

Should $\bigcap_{n = 1}^\infty (a-\frac{1}{n}, b + \frac{1}{n})$ be $(a,b)$ or $[a,b]$

I am confused about the limiting behavior of as $n \to \infty$, $\bigcap_{n = 1}^\infty (a-\frac{1}{n}, b + \frac{1}{n})$. I have read that it is the case that this set becomes closed, but I can't ...
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votes
0answers
50 views

Abstract algebra, Linear algebra 2, or Introduction to Topology? [closed]

Out of these which would you recommend taking over the summer? I'm kind of up in the air about it. Also do you know of any online resources for the following classes?
0
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1answer
39 views

$\mathbb Q$ is totally disconnected. What is the open set in subspace of $\mathbb Q$?

I am trying to understand the proof that $\mathbb Q$ is totally disconnected. If $Y$ is a subspace of $\mathbb Q$ containing two points, $p$ and $q$, we can choose irrational a lying between $p$ and ...
0
votes
2answers
29 views

Show that $x$ is an “accumulation point” of $A$ if and only if every neighborhood of $x$ contains infinitely points of $A$

Let $(X,\Gamma)$ a topological Hausdorff space and $A\subset X$. Show that $x$ is an "accumulation point" of $A$ if and only if every neighborhood of $x$ contains infinitely points of $A$. I ...
3
votes
1answer
77 views

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions?

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions? My attempt to prove this: For contradiction, suppose $X$ is an ...
5
votes
2answers
38 views

Show that $\pi(X, x) = [e_x]$ if $X$ is a finite topological space with the discrete topology.

Show that $\pi(X, x) = [e_x]$ if $X$ is a finite topological space with the discrete topology. I want to show this by showing that there does not exist any path $f$, $\forall x, y \in X$. Assume for ...
1
vote
1answer
76 views

What does it mean for a set to be open in a Topology?

"What does it mean for a set to be open in a Topology?" - I have a hard time getting my head around what it means. Example: Is the set $(0,1]\subset\mathbb{R}$ open in the standard Topology on ...
0
votes
1answer
13 views

Is it true that $U \cap \overline{E_1 \setminus E_2} \neq \varnothing$ implies $U \cap (E_1 \setminus E_2) \neq \varnothing$?

Let $x \in \overline{E_1 \setminus E_2}$. Is it true that if $$U \cap \overline{E_1 \setminus E_2} \neq \varnothing$$ for every open neighborhood of $x$ then $$U \cap (E_1 \setminus E_2) \neq ...