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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57 views

If $(X, \mathcal{D})$ is a discrete space and $(Y, \mathcal{T})$ is any topological space than any $f:X \rightarrow Y$ is continuous

The problem defines $f:(X,\mathcal{D}) \rightarrow (Y,\mathcal{T})$ where $\mathcal{D}$ is a discrete space and $\mathcal{T}$ is any topological space. I have to show that f is continuous. What I ...
5
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1answer
33 views

$X=\{a,b,c\}$ and $\mathcal{T}=\{X, \emptyset, \{a\}, \{b\}, \{a,b\}\}$. Determine if $f:X \rightarrow X$ is $\mathcal{T}-\mathcal{T}$ Continuous

$X=\{a,b,c\}$ and $\mathcal{T}=\{X, \emptyset, \{a\}, \{b\}, \{a,b\}\}$. Assume $f: X \rightarrow X$ is given by $f(a)=a, f(b)=c,$ and $f(c)=b.$ Determine if $f:X \to X$ is ...
0
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1answer
52 views

if $A \subset B \subset X$ then $\mbox{dist}(x,B) \le \mbox{dist}(x,A)$

Prove that if $A \subset B \subset X$ then $\mbox{dist}(x,B) \le \mbox{dist}(x,A)$. My attempt (by contradiction) : Suppose that $\mbox{dist}(x,B) > \mbox{dist}(x,A)$. So we have that: $(\forall ...
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1answer
167 views

Showing a topology is not metrizable

Show $\prod_{N} \mathbb{R}$ with the box topology is not metrizable. The Box Topology on $\prod_{j \in J} X_j$ ($X_j$ topological spaces) is generated by the basis $\left\{\prod_{j \in J} U_j \; ...
3
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2answers
118 views

Relative sizes of Skorokhod and product topologies on space of sample paths

Let $E$ denote a compact metric space. Let $T$ denote the non-negative reals. Let $E^T$ denote the class of all functions from $T$ to $E$. Let $C$ denote the subset of $E^T$ consisting of càdlàg ...
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1answer
42 views

Open set and sequences in $\mathbb{R}$

Let $A \subset \mathbb{R}$. Prove that $A$ is an open set if, and only if, the following condition is satisfied: " if a sequence $(x_n)$ converges for a point $a \in A$, then $x_n \in A$ for all $n$ ...
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1answer
36 views

Density of the function class

Let $X$ be any set and let $[0,1]^X$ (the class of all functions $X\to[0,1]$) be endowed with the metric given by $\rho(f,g):=\sup_{x\in X}|f(x) - g(x)|$. Consider any class of functions $\mathscr ...
2
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2answers
273 views

Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable.

Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable. $\Bbb{R}$ must be Hausdorff. For $x_1, x_2 \in \Bbb{R}$ (where $x_1 \not= x_2$), if $d$ ...
3
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2answers
288 views

Infinite disjoint class of open subsets in an infinite Hausdorff space.

Let X be an infinite Hausdorff space. Prove that there exist an infinite disjoint class of open subsets of X. Ok the first time I tried to prove this I started by taking pairwise disjoint sets given ...
2
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1answer
65 views

Another question on second countable spaces

Let $X$ have countable chain condition and point countable base. Is $X$ second countable? I thing it don't need to be. However I have no examples at hand. Thanks for your help.
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2answers
45 views

Show that $\bigcup _{\alpha \in J}\operatorname{cl} A_\alpha \subseteq\operatorname{cl}\bigcup _{\alpha \in J} {A_\alpha} $

$\newcommand{\cl}{\operatorname{cl}}$Let $\{A_\alpha\}_{\alpha\in J}$ be a collection of subsets of a topological space X. Show that $\bigcup _{\alpha \in J}\cl A_\alpha\subseteq\cl\bigcup _{\alpha ...
5
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2answers
107 views

Applications of showing a set is both open and closed?

A general technique is as follows: To show that a property holds for a connected space, one can prove that the set of all points that satisfy this property is nonempty and forms a closed and open ...
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2answers
88 views

What is a 0-ball?

I'm reading a paper that says $\bigcap V_{T,X}$ is either empty or a closed $l$-ball where $T \subset S$ is a subset of points $S$ and $\operatorname{card}{T} = m + 1 - l$ where $m$ is the dimension ...
0
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1answer
75 views

A = (0, 1/2). Find the closure of A in X = (0,1].

$A = (0, 1/2)$. Find the closure of $A$ in $X = (0,1]$. So $X = (0,1]$ is a topological space with subspace topology $T' = \{ U \bigcap (0,1] \mid U\text{ is open in }\mathbb{R}\}$. The basis for ...
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1answer
62 views

Prove that $\bar{h}:Y\to Z$ is a continuous function.

I've run into this rather tricky question (to me at least). Let $(X,\mathcal{T})$ be a topological space and let $Y$ be another set and let $f:X \to Y$ be a surjective function, and equip $Y$ with ...
3
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2answers
164 views

Why it does not produce a Klein bottle?

I cannot understand why the action $\mu : (\mathbb{Z}\oplus \mathbb{Z})\times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 $ given by $\mu((m,n), (x, y)) = (x+ m, (-1)^m(y + n))$ does not produce the ...
2
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2answers
209 views

How to construct a contractible space but not locally path connected?

I am looking for a space which is contractible and not locally path connected. I know the cone $CX$ of every space $X$ is contractible. Besides, it seems that if $X$ is locally path connected, so is ...
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2answers
68 views

Proving that $\Bbb{R}_\ell$ is finer than $\Bbb{R}$.

Let us take the two topologies $\Bbb{R}_\ell$ and $\Bbb{R}$. The book "General Topology" by Munkres says that $\Bbb{R}_\ell$ is finer than $\Bbb{R}$. This article says that every open set of $\Bbb{R}$ ...
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3answers
155 views

Is the set of extended real-valued numbers open or closed

If I assume that my topology is defined on the extended real-valued numbers, then $\mathbb{R}\cup\left\{-\infty,+\infty\right\}=\left[-\infty,+\infty\right]$, acting as my entire space, is both open ...
0
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2answers
58 views

Is this a valid proof that $\mathbb{R}$ is connected?

Suppose $\mathbb{R}$ is not connected, i.e., $\mathbb{R}=A \cup B$, where $A,B$ are open sets that are disjoint. $A$ is bounded above by each element of $B$, so $A$ must have a supremum, call it $x$. ...
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2answers
600 views

Difference between interior and set of accumulation points

I don't understand the difference between the interior of a set, and the set of all its accumulation points. My understanding of an accumulation point is any point in a set which has an epsilon ...
3
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1answer
72 views

$A$ is uncountable $\implies$ $A'$ is uncountable?

For $A⊆\mathbb R$ , let $A'$ denote the set of all limit points of $A$ . If $A$ is uncountable , then does it necessarily mean that $A'$ is also uncountable ?
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1answer
226 views

Proof: A connected metric space which contains more than 1 point is never countable. [duplicate]

This is an exercise in Munkres's book of topology. If $X$ is a connected metric space and there are at least two points in $X$, then $X$ is not countable. I have attempted to find the proof by ...
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2answers
70 views

A question on second countable space

A family $\mathcal U$ of subsets of a space $X$ is called k-in-countable if every set $A \subset X$ with $|A|=k$ is contained in at most countably many elements of $\mathcal U$. If $X$ is a ...
0
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2answers
76 views

Prove that a continuous $f$ in $(0,1)$ can be extended into its one-point compactification if the limit at both end point exist and equal

Let $X = (0,1)$. Consider the one-point compactification of $X$ (which is homeomorphism to $S^{1}$). Prove that a bounded continuous function $f:(0,1) \rightarrow R$ is extendable to this ...
3
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2answers
295 views

Questions on positive definite matrices

First, in this discussion, I am only considering real matrices. Second, I have a few questions I am ruminating on related to symmetric matrices. Some of these questions I need someone to say my ...
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1answer
74 views

Baire Category Theorem in a Smooth Manifold

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$. How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using Baire category theorem? and which of the theorem version is ...
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1answer
719 views

Let X be any uncountable set with the cofinite topology. Answer the three questions:

Let X be any uncountable set with the cofinite topology. Is this space 1st countable? I don't think this space is 1st countable because it seems that there must be an uncountable number of ...
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2answers
218 views

Need an example of a space which is not first countable

Give an example of a space which is NOT first countable & in which every singleton set is : $ G_\delta $ . I have just found out one example where the space is NOT first countable is: any ...
2
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2answers
234 views

Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous

$$d_\infty = \max|x_i - y_i|$$ $$d_1 = \sum_{i=1}^n |x_i - y_i|$$ The first part of this question was to prove that the identity map $$(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$$ is continuous, ...
0
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1answer
56 views

Connectedness of subsets

Let $(X,d_x)$ be a metric space, and let $A$ be a non connected, non empty, closed subset of $X$. Can I conclude that $A$ equals to $X$ and $X$ is not connected? It seems that I had wrong ...
1
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1answer
55 views

If $F_1 \cup F_2 = \mathbb{R}$ and $F_1,F_2$ are closed sets then $\mbox{Int}F_1 \neq \emptyset$ or $\mbox{Int}F_2 \neq \emptyset$

Prove that if $F_1 \cup F_2 = \mathbb{R}$ and $F_1,F_2$ are closed sets (in euclidean space) then $\mbox{Int}F_1 \neq \emptyset$ or $\mbox{Int}F_2 \neq \emptyset$ My idea is prove that by ...
2
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2answers
553 views

Show that the topological space ( X, $\tau$ ) is not metrizable

For the topological space ( X, $\tau$ ), with X = {0, 1} and $\tau$ = { $\emptyset$ , {0}, {0,1} } , prove that ( X, $\tau$ ) is not metrizable. I know intuitively it can't be but don't know how to ...
1
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1answer
30 views

Where I can find reference on Katetov's extension $kN$ of the natural numbers?

I’m looking for references on Katetov's extension $kN$ of the natural numbers? However I cannot find it. Is this separable and is this a countable union of closed discrete subspace of it? Thanks ...
2
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4answers
117 views

Is $\left\{ \frac{1}{n}: n \in \mathbb{N} \right\} \cup \left\{ 0\right\}$ closed set? [duplicate]

Is it true that $\left\{ \frac{1}{n}: n \in \mathbb{N} \right\} \cup \left\{ 0\right\}$ is closed set? I suppose that yes, but I have no idea how can I prove it.
0
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1answer
31 views

Questions on covering space

Following is a paragraph of a paper I am reading: But I cannot understand this image, maybe it is because I have no idea about coverings. Could anyone explain it to me? Particularly, you could just ...
0
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2answers
56 views

what is $X / \cong $ ?? where $\cong $ is given by

what is $X / \cong $ ?? Suppose $X = \mathbb{R}^2 $. and we define $$ (x_1,y_1) \cong (x_2, y_2) \iff x_1 + y_1 =x_2 + y_2 $$ With this equivalence relation, we get that the partition is the ...
4
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0answers
117 views

Topologist Sine Curve

I am trying to prove that the topologist sine curve is not path connected. I think I have a proof but my proof relies on Intermediate Value Theorem. So, I was wondering if there is a way to prove it ...
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1answer
221 views

The intersection of open intervals.

For $i=1,2, \cdots, n$, let $I_{i}=(a_i, b_i)$ be an open interval. Show that $\cap_{i=1}^n I_i$ is either the empty set or an open interval. Can anyone show me how to do this because this is slightly ...
0
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4answers
172 views

A base for a topology

I am quite confused about what exactly a base for a topology is. I understand it when the topologies are pretty simple, but things start to get a little confusing for me after awhile. For example, ...
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2answers
49 views

One question about topology.

Why is the set $ A=\{ (x ,x^{-1}):0<x\leqslant 1\}$ is closed in $\Bbb R^2$ but is not bounded? Why is the set $ S=\{(x,\sin(x^{-1})) :0<x\leqslant 1\}$ is bounded in $\Bbb R^2$ but is not ...
4
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1answer
369 views

Non-Euclidean Space in Dungeons and Dragons

In Dungeons and Dragons, the world is mapped out into five-foot squares. Spheres are represented as cubes, and cones look really weird. However, straight lines remain straight, and a rectangular room ...
3
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2answers
160 views

What is compactification generally?

In wikipedia, compactification is defined as an topological imbedding $f:X\rightarrow Y$ such that $f(X)$ is dense in $Y$. However, Munkres-Topology requires $Y$ to be Hausdorff to be called a ...
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1answer
30 views

question about cluster points of a set of cluster points

Let A be a set in a metric space X. A′ is the set of cluster points of A. Is it A′′ ⊆ A′ ? I think it is not. But I can not give a counterexample or proof. Could someone give me a clue? Thank you!
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2answers
51 views

convexity and the interior sphere condition

Consider $\Omega $ a open, convex bounded subset of $R^n$. Let $x_0 \in \partial \Omega$. I believe that exists a open ball $B \subset \Omega$ such that $\partial B \cap \partial \Omega = \{ x_0 \}$. ...
0
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1answer
279 views

Show f is continuous if and only if for any x $\in$ X and any open set O$^y$ in Y containing f(x), …

Suppose $X$ and $Y$ are topological spaces with topology $T^x$ and $T^y$ Let $f: X \rightarrow Y$ be a function. Show $f$ is continuous if and only if for any $x \in X$ and any open set $O^y$ in ...
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2answers
956 views

Show that f is a continuous function if and only if for every closed set C in Y, f$^{-1}$(C) is closed in X.

Suppose X and Y are topological spaces with topology $T^x$ and $T^y$ Let f: X $\to$ Y be a function. Show that f is a continuous function if and only if for every closed set C in Y, f$^{-1}$(C) is ...
0
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1answer
44 views

Countability in topology

I am looking at the following example: $\mathbb{R}$ is second countable because consider the open intervals $(a,b)$ with rational endpoints. This is countable base for the usual topology on real line ...
2
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1answer
42 views

Is $f: \mathbb{R}_l \to \mathbb{R}, f(x) = 1$ for $x\geq 0$ and $f(x) = -1$ for $x < 0$ continuous?

Let $\mathbb{R}$ be the set of real numbers with standard topology. Let $\mathbb{R}_l$ is the set of real numbers with lower limit topology. Is $f: \mathbb{R}_l$ $\to \mathbb{R} $ given by $$ f(x) ...
2
votes
2answers
286 views

Genus of a curve: topology vs algebraic geometry

In topology one defines the genus $g$ of a connected orientable topological manifold $X$ as: The maximum number $g$ of cuttings along non-intersecting closed simple curves without rendering the ...