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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given a basic neighborhood space, finding an algorithm that will determine the open sets of that space.

I would like to figure out how to find open sets of the basic neighborhood space ($\mathcal{B}_X,X$) where $X$ is the space and $\mathcal{B}_X$ is the basic neighborhood on $X$. The second to last ...
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1answer
223 views

Shortcut in proof of continuity/differentiability in inverse function theorem

The messiest, least interesting part of the various proofs of the inverse function theorem comes after you have constructed the inverse function and must now establish continuity and ...
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52 views

A special filter on cartesian product of sets

The following is inspired by this article in nLab (in attempt to simplify it using my notions of funcoids and reloids, which notation is however outside of the scope of this question). Fix a set $U$. ...
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133 views

A generalization of the generalized tube lemma

I am trying to prove the following generalization of the generalized tube lemma: Let $\{X_t\}_{t \in T}$ be a family of Hausdorff spaces and $\prod_{t \in T}A_t$ be a compact subset of $X=\prod_{t ...
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1answer
85 views

How can the sum of two closed cones be not closed?

Can there be two closed cones $K_1$ and $K_2$ in $\mathbb{R}^3$ such that $K_1+K_2$ need not be closed?
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90 views

How can the y-axis in $\mathbb{R^2}$ be open?

I have read that $\{(x, \frac{1}{x}): x \neq 0\}$ is closed in $\mathbb{R^2}$. So hence the complement of this set, $\{x = 0\}$, i.e. the y-axis must be open? But we cannot put an open ball with ...
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91 views

Prove that a topological space $(X, \tau)$ is $T_2$ if and only if the diagonal $D=\{(x,x):x \in X\}$ is closed subset of $X\times X$ [duplicate]

Prove that a topological space $(X, \tau)$ is $T_2$ if and only if the diagonal $D=\{(x,x):x \in X\}$ is closed subset of the product space $X\times X$ => assume that $(X, \tau)$ is $T_2$, I know ...
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93 views

Proof that the map is an open map.

We were asked to prove that the map $f\ :\mathbb{R}^{n+1}/{0}\longrightarrow S^{n}$ given by $f(x)\ =\ x/||x||$ is an open map. My approach :- it is enough to show that for any $x\in \mathbb{R}^{n+1}$ ...
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3answers
57 views

Let $S\subset\mathbb{R}$ bounded Prove that $\mbox{diam}\left(S\right)=\mbox{diam}\left(\overline{S}\right)$

Let $S\subset\mathbb{R}$ bounded and $\mbox{diam}\left(S\right)=\sup\left\{ |x-y|:x,y\in S\right\} $ Prove that $\mbox{diam}\left(S\right)=\mbox{diam}\left(\overline{S}\right)$ I have trouble with ...
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43 views

How to deform a curve in specific manner

I am wondering whether we can deform a path in specific ways continuously i mean if there is a closed piece wise $C^1$ smooth path which has to be deformed to another piece wise $C^1$ smooth path. Let ...
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78 views

Show $\Omega$ is simply connected if every harmonic function has a conjugate

Prove: If every harmonic function on $\Omega$ has a harmonic conjugate on $\Omega$, then $\Omega$ is simply connected. Same question is asked here but no proof is presented: is the converse true: in ...
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47 views

Why is convergence in measure topologizable?

I'm aware that pointwise convergence and uniform convergence are topologizable since the former can be made by seminorms and the latter with a norm. I'm also aware that pointwise a.e. fails because ...
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1answer
58 views

Suppose $A \subset S$ and both $A$ and $S$ are regular surfaces. Show that $A$ is open in $S$

Suppose $A \subset S$ and both $A$ and $S$ are regular surfaces. Show that $A$ is open in $S$ (w/ respect to subspace topology on $\mathbb{R}^3$. Note that the definition of a regular surface $S$ is ...
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77 views

In what metric spaces is a closed and bounded set compact?

Is there a characterization of a metric space $X$ such that for every $A\subseteq X$, $A$ is compact iff $A$ is closed and bounded? Something that generalizes $\mathbb R^n$?
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16 views

Showing two Final Topologies are the Same

Let $X$ be a topological space. We denote by $\tau_\mathbb{R}$ the final topology induced by the family of continuous maps $\varphi:\mathbb{R}\rightarrow X$, and by $\tau_I$ the final topology ...
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1answer
66 views

What are the formulas for topological transformations? How to obtain them?

I'm reading Flegg's From Geometry to Topology, the author says that in Euclidean geometry, translation and rotation are: $$T:(x,y)\to(x+a,y+b)$$ $$R:(x,y)\to(x \cos \phi - y \sin \phi, x \sin \phi +y ...
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1answer
32 views

Can anything be said for the topology of a topological monoid?

A topological group is one in which the group operations (the multiplication and inverse) are continuous, or equivalently as a group object in $\mathbf{Top}$. They are uniformisable and hence are ...
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45 views

Creating continuous functions - topology

I am having some difficulty in understanding the following proof. -- If $X$ and $Y$ is a topological space, and there is a continuous function $f:X \to Y$. Now if $Z$ is a subspace of $Y$ ...
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1answer
39 views

Is there a topological characterisation of non-Archimedean local fields?

A local field is a locally compact field with a non-discrete topology. They classify as: Archimedean, Char=0 : The Real line, or the Complex plane Non-Archimedean, Char=0: Finite extensions of the ...
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51 views

caracterization of lower semicontinuous functions

Let $X$ a topological space satisfying the first contability axiom. I want to prove the following result: $\varphi : X \rightarrow R $ is lower semicontinuous (this mean that $\varphi^{-1} (a, + ...
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1answer
48 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
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3answers
358 views

Inverse limit of an inverse system of topological spaces

Given an inverse system $\mathcal G=\{X_i\}$ of topological spaces over some directed set $I$. If $X=\prod\limits_{i\in I}X_i$, the inverse limit $X^*=\varprojlim X_i$ of $\mathcal G$ is a subspace ...
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2answers
124 views

Are a finite cylinder and the corresponding planes iso/homeomorphic?

Let me give some context first. In the scope of physics, I often have to compute the area of the side of a right circular cylinder with height $h$ and radius $r$, namely $2\pi rh$. I think this can ...
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90 views

How to show that this function is continuous at the point x=0, but not continuous elsewhere with the topological definition?

for the function $f:\Bbb R\to\Bbb R$ (standard topologies) defined by $f(x)= x$ if $x$ is a rational number $f(x)= -x$ if $x$ is not rational Prove that $f$ is continuous at the point $x=0$, but ...
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1answer
183 views

Definition of a nowhere dense set

I'm currently studying metric spaces through Gamelin and Greene's Introduction to Topology. While studying about completeness I got stuck with this concept of nowhere dense subset. The book defines a ...
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31 views

Cluster Points of a Convergence Space

I'm trying to find the characteristic properties (axioms) of cluster points in a convergence space. I've come up with a minimal two: (let $\mathrm{adh}(\mathcal{F})$ be the set of cluster points of ...
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1answer
44 views

Find a subbasis for metric topology of $\mathbb R^2$ such that it's not a basis itself

Find a subbasis for metric topology of $\mathbb R^2$ such that it's not a basis itself . I know that metric topology is the collection of open balls in $\mathbb R^2$ I also know how to use the ...
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1answer
57 views

Is there a name for a topological space $X$ in which Every closed subset $A\subsetneq X$ is contained in a countable union of compact sets

As was recommended for me in here I would like to share the following question with you: Is there a name for a topological space $X$ which satisfies the following condition: Every closed subset ...
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22 views

What can we say about the space just by looking at its Borel sets?

What can we say about a compact space $X$ just by looking at the Borel sets of $X$? In general, it seems that not much but maybe it is still not a bad question. For instance, let $X$ be a compact ...
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72 views

Limit Point vs Boundary Point

I'm reading Kosniowski's book on algebraic topology, and I have a question about how he defines limit points. He says that for a subset $Y$ of a topological space $X$, the limit points of $Y$ are ...
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1answer
60 views

Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is contained in a countable union of compact sets Does Baire space satisfy this ...
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1answer
142 views

What is the definition of a product topology?

I am new to advanced mathematics and I recently started reading a book on topology. I am struggling to understand what it is saying in this paragraph. This is what it says: Let $E_i$ ...
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1answer
320 views

The Line with two origins

I have seen descriptions of the "line with two origins" using quotient spaces. My professor has defined it in an alternate way. However, I can not wrap my head around how the following descriptions ...
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29 views

$(0,1)^\omega$ homeomorohic to $R^\omega$?

Since $(0,1)$ is homeomorphic to $R$ and an infinite product of homeomorphisms is a homeomorphisn?
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1answer
77 views

Is Baire space $\sigma$-compact?

Is Baire space $\sigma$-compact? Thank you!
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2answers
91 views

Cell Complex: Proposition 5.5 in John Lee's book “Introduction to Topological Manifolds”

The proposition reads: "Suppose $X$ is an $n$-dimensional CW complex. Then every $n$-cell of $X$ is an open subset of $X$." The proof first shows that the intersection of any $n$-cell of $X$ with the ...
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119 views

Is $\mathbb R^{\omega}$ homeomorphic to $\mathbb R^{\omega} \times \mathbb R^{\omega}$?

As a study exercise, I'm trying to find a topological space $X$ which is homeomorphic to $X \times X$. I began thinking of simple examples involving $\mathbb R$ but then realized my best bet would be ...
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1answer
30 views

Given tangent space of submanifold of Lie group, is it possible to recover the submanifold?

I have computed the tangent space of certain submanifolds (the unstable manifolds) of a Lie group at a particular point. I know that the exponential map lets us move between the Lie algebra and the ...
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1answer
87 views

Question on Baire Property

In reading Banach's book, Theory of Linear Operations, I have a question on the definition of Baire property, or Baire condition in Banach's book. Here is the definition in Banach's book: Definition. ...
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1answer
80 views

Topological Spaces, Homeomorphism

Let $X,Y,U,V$ be topological spaces, and $X$ is homeomorphic to $U$ and $Y$ is homeomorphic to $V$ Then is $X\times Y$ homemorphic to $U\times V$? I have got the following maps $n\colon X ...
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1answer
323 views

The implications of Completeness and the Continuity axiom for utility representation

Completenes means that every basket of goods in some set previously defined is comparable with the use of a complete preference. Now, with the additional assumption that the preferences are ...
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51 views

topological vector space of measure functions

Let $(X, \mathcal X, \mu )$ be a measure space, and let $ L(X)$ be the space of measurable functions $f: X \to \mathbb C$. Show that the sets $B(f, \epsilon ,r ): = \{ g \in L(X) : \mu( \{ x : | f(x) ...
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Is an infinite line the same thing as an infinite circle?

Imagine that you are sitting next to a line that extends infinitely in both directions. Is it possible to distinguish it from an infinite circle? From my poor understanding of topology, I would ...
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55 views

About Hilbert spaces

How can I prove this fact: We're working in a Hilbert space $$ \mathcal{H} := \left\{ (x_n)_{n \in \mathbb{N}} \in {\mathbb{R}}^{\mathbb{N}} \mid \sum_{n=1}^{\infty}\,(x_n)^2 < \infty \right\} $$ ...
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1answer
40 views

An open cover $\{U_\alpha\}$ of $X$ is locally finite iff each $U\alpha$ intersects $U_\beta$ for only finitely many $\beta$

I am trying to prove: Lee, Smooth Manifolds, Exercise 2.9. Show that an open cover $\{U_\alpha\}$ of $X$ is locally finite if and only if each $U\alpha$ intersects $U_\beta$ for only finitely many ...
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113 views

Topology inducing Order

We know by Munkres that any (total) ordering induces a topology and (luckily) for the real line that coincides with the euclidean topology. In fact, this construction can be carried over to any ...
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3answers
96 views

Limits of Topological Vector Spaces

Let $X, Y_1, Y_2, \cdots$ be a sequence of topological vector spaces, and let $f_n : X \to Y_n$ be a sequence of continuous linear maps. Define the product space $\mathcal Y_N := Y_1 \times \cdots ...
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4answers
157 views

Let $a_n$ be a convergent sequence wit limit $L$. With out using the Heine-Borel theorem, prove that the set $\{L,a_1,a_2,…\}$ is compact.

Let $a_n$ be a convergent sequence wit limit $L$. With out using the Heine-Borel theorem, prove that the set $\{L,a_1,a_2,...\}$ is compact. I know that $a_n$ is convergent to $L$, meaning for all ...
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53 views

Topological contraction on compact spaces

This is a follow up question. You can see the original here. I have the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a ...
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39 views

A question about rectifiable curves

Does every rectifiable curve that is a subset of the Euclidean plane have zero two-dimensional Lebesgue measure?