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
860 views

How does homeomorphism map sets boundaries?

I'm at the end of my first course on general topology, but this topic was not well developed. I can tell that an homeomorphism preserves the quality of a point to be a boundary point for a subset of a ...
2
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1answer
338 views

Comparison between various types of cell complexes

There are the following (and more) types of geometric cell complexes: 1) The geometric realization of a simplicial set 2) CW-complexes 3) The geometric realization of an abstract simplicial complex ...
2
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1answer
3k views

Defining a Limit Point of A Set

Limit Point is defined as: Wolfram MathWorld: A number $x$ such that for all $\epsilon \gt 0$, there exists a member of the set $y$ different from $x$ such that $|y-x| \lt \epsilon$. Proof Wiki: ...
2
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1answer
353 views

On compact space with order topology

We are familar with that for the first uncountable cardinality $\omega_1$, the topological space $[0,\omega_1]$ is compact. I find the proof for the $\omega_1$, is also for every regular cardinality. ...
2
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1answer
532 views

A universal property for the subspace topology

Let $X$ be topological space and $Y$ be a subset of $X$ with $i\colon Y\to X$ the inclusion map. Show that the induced topology of $Y$ is characterized by the following property: A function $f\colon Z ...
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2answers
1k views

Topology for beginners [duplicate]

Possible Duplicate: best book for topology? Please Suggest some good books on Topology and Functional Analysis. It would be good if somebody can post links of video lectures related to these. ...
2
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1answer
103 views

Is a valuation a continuous map?

What I want to know is: the preimage of an integer by a valuation map is an open set? Otherwise: Can we cover a valuation field by open sets with elements with fixed valuation?
2
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2answers
72 views

what does linear type mean?

What does it mean when we say a topological group $\Gamma$ has linear type? Is it an algebraic property or a topology property? I wonder if anyone could give some references.
2
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1answer
275 views

WOT closure and SOT closure of convex sets

I am reading some papers on operators acting on Banach spaces and one of them uses the following fact: If a vector space has two locally convex topologies with identical collections of continuos ...
2
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1answer
902 views

How to show multiplication and inversion are continuous in this topology on $C(X)$?

The sets$$\big\{ \{f\in C(X) : |g-f| \le u \} \;\big\vert\; g\in C(x) \text{ and } u \text{ is a positive unit of } C(X)\big\}$$ form a base for some topology on $C(X)$. Corresponding to this topology,...
2
votes
1answer
273 views

Product space and product topology

I have been told that (in context of Tychonoff's theorem) that $\prod_{i\in I}X_i$ (take for example $I=\{1,2\}$) and $X_1 \times X_2$ are isomorphic. Generally when $I$ is countably infinite or ...
2
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3answers
472 views

In a first-countable space, an accumulation point of the set of terms in a sequence is also a limit-point of the sequence

I ran into something which seems like it should be true (and must be true because it is used in the proof of a theorem). I've extracted the detail that I cannot quite verify. Let $(x_{n})$ be a ...
2
votes
2answers
110 views

Let $S\subseteq\mathbb{R}^2$ be the set of points where $f$ is continuous - is $S$ open or closed? Is it empty?

Define function $f : \mathbb{R^2}\rightarrow \mathbb{R}$ by $$f(x, y) = \begin{cases}1,&\text{if }xy=0\\2,&\text{otherwise}\;.\end{cases}$$ If $$S = \{(x, y): f\text{ is continuous at the ...
2
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1answer
261 views

Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number

Let $f$ be analytic on the closed unit disk centered at the origin and $|f(z)| < 1$ for $|z| = 1$. Show that $f$ has exactly one fixed point inside the open unit disk. That is, there exists a ...
2
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1answer
330 views

What is the definition of a cross cap?

I have looked this up in several places but can only find heuristic descriptions. What I'm after is a definition like If X has certain properties, the cross cap on (of?) X is the quotient of Y with Z....
2
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1answer
132 views

Various types of TQFTs

I am interested in topological quantum field theory (TQFT). It seems that there are many types of TQFTs. The first book I pick up is "Quantum invariants of knots and 3-manifolds" by Turaev. But it ...
2
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4answers
110 views

Is this space Pathconnected?

Is the infinite unions of $\cup_{n=1}^{\infty} (\frac{1}{n},1) \cup\{{0}\}$ path connected. Are they actually equal to the space $[0,1)$? Thanks
2
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1answer
150 views

The set of linear fractional transformations that preserves the open unit disk is equicontinuous on every compact subset in it

Let $\Delta(a, r)$ be the open disk of radius $r$ centered at the point $a$ in the complex plane, and $\operatorname{Aut}(\Delta(0, 1))$ be the set of linear fractional transformations that preserves ...
2
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1answer
59 views

$X \setminus cl(x)$ meet-irreducible in every T$_{0}$ space X

I'm reading the paper 'Frames' by A. Pultr and I'm having trouble proving the following: Let $\Omega(X)$ be the system of all open sets of X. This is a complete lattice. Let now X be a T$_{0}$ space, ...
2
votes
1answer
84 views

Is this a closed subset?

Is the infinite union of intervals $(\frac{1}{n}, 1]$ where $n \in \mathbb{N}$ i.e. $\bigcup\limits_{n=1}^\infty (\frac{1}{n}, 1]$ a closed subset of $\mathbb{R}$?
2
votes
1answer
309 views

Closure, compactness and connectedness of a set

Question: What is the closure of the set $\{(-1/2)^n \, : \, n \in \mathbb{N} \} \cup \{0\}$ as a subset of $\mathbb{R}$? Is it compact? Is it connected? My answer: The closure is the set itself. It ...
2
votes
1answer
101 views

subset in $\mathbb{R}^2$ with the Euclidean metric is closed and whether it is complete

Determine whether the given subset in $\mathbb{R}^2$ with the Euclidean metric is closed and whether it is complete? $S^1=\{(x_1,x_2) \in \mathbb{R}^2\mid x_1^2 + x_2^2 =1\}$; $B_1=\{(x_1,x_2) \in\...
2
votes
1answer
395 views

Definition of a Cauchy sequence in terms of a Cauchy filter

Prerequisites in case you may need and I am correct about them: An entourage is a member of uniformity structure on a set for it to be a uniform space. Intuitively, an entourage is a relation on a ...
2
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1answer
652 views

Examples of Topological Embeddings and Quotient Maps

What is an example of a topological embedding that has no continuous left inverse? a quotient map that has no continuous right inverse?
2
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1answer
583 views

Explicit description in set-builder notation of an arbitrary open set of the product topology

Short version: Is it possible to explicitly describe the open sets of the product topology (of arbitrary topological spaces) via set-builder notation? (Or differently formulated: What do to if set set ...
2
votes
1answer
229 views

Is there a topological space which is star compact but not star countable?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. A ...
2
votes
1answer
148 views

Closure of Topological Spaces

Let $A$ be a subset of a topological space $X$ and let $B$ be a subset of $A$. I need to prove that the $\mathrm{Cl}(A)-\mathrm{Cl}(B)$ is contained in $\mathrm{Cl}(A-B)$. $\mathrm{Cl}$ denotes the ...
2
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1answer
146 views

$\{0,1\}^\lambda$ and its clopen subsets

Let $\lambda$ be an uncountable cardinal and let $X=\{0,1\}^\lambda$ be endowed with the product topology ( $\{0,1\}$ is discrete). Is there an uncountable chain (with respect to inclusion) of clopen ...
2
votes
1answer
220 views

About a finite chain of connected sets.

Suppose $\mathscr{A}$ is a collection of connected subsets in some topological space such that for any two $A$, $B$ in $\mathscr{A}$, there are $A=C_0,C_1,\dots,C_n=B$ such that $C_i$ and $C_{i+1}$ ...
2
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1answer
124 views

Continuous functions on $\omega_1$

Let $\lambda<\omega_1$ be a limit ordinal and consider the set $X=\{\alpha+1\colon \alpha<\lambda\mbox{ is a limit ordinal}\}$ Is the characteristic function $\chi_X$ continuous on $\omega_1$ ...
2
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1answer
345 views

Stone–Čech compactification with topologist's sine curve

The Stone–Čech compactification $\beta((0, 1))$ of $(0, 1)$ can be described as the closure of $(0,1)$ in $\prod_{j\in J} \mathbb{R}$ where $J$ is the set of bounded continuous functions $f : (0,1) \...
2
votes
2answers
256 views

Sequential continuity for quotient spaces

Sequential continuity is equivalent to continuity in a first countable space $X$. Look at the quotient projection $g:X\to Y$ to the space of equivalence classes of an equivalence relation with the ...
2
votes
1answer
99 views

Action of Homeomorphism on Boundary of Subspace

Is the following true? Suppose $A\subset X$ is closed and $H\colon X\to Y$ is a homeomorphism. Then $\partial H(A) = H(\partial A)$.
2
votes
2answers
413 views

Uniqueness of the Quotient Topology

Let $q:X\to Y$ be a surjective map, where $(X,\tau_X)$ is a topological space. The quotient topology $\tau_q$ on $Y$ is given as $U\in \tau_q$ iff $q^{-1}(U)\in \tau_X$. Suppose that there is ...
2
votes
1answer
138 views

Is this weird looking topology on a subset of the rational plane $T_2$?

I'm investigating a topology on that rational plane $\mathbb{Q}^2$, with a subbase which I've had a hard time getting my hands on. Suppose $X=\{(x,y)\in\mathbb{Q}^2\mid y\geq 0\}$ be the given subset,...
2
votes
2answers
180 views

Interior points of correspondence

Let $\Gamma(x)$ be a correspondence (i.e. a set-valued function) between two Euclidean spaces which is continuous (i.e. both lower- and upper-hemicontinuous). If $y$ is a point in the interior of $\...
2
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2answers
236 views

Can anyone tell me why the arclength integral is an uppersemicontinuous function on the set of continuously differentiable real-valued functions?

The continuously differentiable functions are equipped with the topology induced by the sup norm. I know that I can make the arclength integral close to the arclength of a piecewise linear function. ...
2
votes
3answers
168 views

The continuity of a function in the uniform topology

I am doing the exercises in the book Topology(2nd edition) by Munkres. Here is my question(page 127, question 4(a)): Let $h:R\to R^\omega$ be a function defined by $h(t)=(t, t/2, t/3, \ldots)$ where ...
2
votes
1answer
255 views

If $X$ is regular then so is $C(X,X)$

I can't find a proof of this result, can anyone help me? Let $X$ be a topological space and $C(X,X)$ the space of all continuous functions from $X$ to itself. Suppose $X$ is regular, then $C(X,X)$, ...
2
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1answer
145 views

Heine-Borel - Finite Intersection equivalence in subspaces

I'm doing a course in Calculus/Real Analysis, and we're covering really basic topological ideas (metric spaces, open/closed sets etc.). From my notes it looks like we proved the following in class (...
2
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2answers
139 views

Countable subfamily with union equal to that of the containing family

Recently it was explained here that in a second countable topological space $X$, any base admits a countable subfamily which is also a base. I know a base $\mathcal{B}$ covers the space $X$, so $\...
2
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1answer
559 views

Continuous vector field on surface

On plane we can easy to define continuous vector field: $$\mathbb{R}^2\ni x\longmapsto v(x)\in\mathbb{R}^2,$$ where real-valued function $v_1(x)$ and $v_2(x)$ are continuous. But when we try to ...
2
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4answers
589 views

A function is not continuous, but the image of convergent sequences converge

Building off a previous question, I'm trying to prove some properties about a certain function, but I may be flubbing the whole thing. Suppose I have a well-ordered set $(B,\leq)$ where there is a ...
2
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1answer
776 views

The mapping cylinder of CW complex

If $X,Y$ are CW complexes and $f$ a cellular map from $X$ to $Y$, is it true that $M_f$ is a CW complex?
2
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1answer
860 views

Open sets in product topology

I'm quite certain that this should be trivially simple, but it's very late and I'm not that bright at the best of times: $\{(X_\lambda, \mathcal{U}_\lambda)\,|\,\lambda \in \Lambda\}$ is a family of ...
2
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2answers
1k views

Quotient space of the torus with two points identified

Take the torus $T=S_1 \times S_1$. Choose two points $x,y \in T$ and define a quotient topology by identifying $x$ and $y$. Let $X$ denote the quotient space. Prove that: a) Compute the fundamental ...
2
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1answer
185 views

strong separation of sets

I'm going through some of the practice questions in my textbook, and one has me stumped, on account of seeming almost too straight-forward. We're instructed to show that two sets, $A$ and $B$ of $\...
2
votes
2answers
310 views

closed set in another closed set

I'm new here and I'm hoping that maybe I could get some help with something my teacher told me. He said that it is possible to have a closed set nested within another closed set where the intersection ...
2
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1answer
213 views

Path connected attaching map

I am trying to showed that if $X$ and $Y$ are path connected then $X \coprod_f Y$ is path connected. (The adjunction space). Let $A \subset X$ , and let $f:A \to Y$ be the attaching map. (Note: ...
2
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1answer
105 views

Extension to a continuous function of a set of specified values

I was able to prove an extension of the Ham Sandwich Theorem (namely; that given any 3 integrable functions on $\mathbb{R}^3$, there exists a plane which simultaneously divides their total integrals ...