Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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1answer
11 views

How to show that a regular space with non-countable basis is not normal

I read that every regular space $X$ with a countable basis is normal. For if $A$ and $B$ are disjoint closed sets in $X$, one can form a neighborhood $U=\bigcup U_j$ over $A$ (from countable bases $...
2
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1answer
33 views

Approximation of irrationals in $[0,1]$

Enumerate the rationals in $[0,1]$ as follows: $\{0,1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5,... \}$, and let $F_n$ be the set consisting of the first $n$ elements of this enumeration. Is there a sequence ...
2
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3answers
28 views

Is an infinite set with no limit point unbounded in an arbitrary metric space?

Given an infinite set $X$ with no limit points, is $X$ unbounded? (In an arbitrary metric space) I only know how to do this in $\mathbb{R}^k$. Since $X$ has no limit points, $X$ is closed. An ...
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1answer
26 views

Rudin RCA 5.6-7 Forms of Baire’s Theorem

In Rudin's Real and Complex Analysis, he mentions two (equivalent) forms of Baire's theorem, $(1)$: for a complete metric space $X$, the intersection of every countable collection of dense open ...
2
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1answer
57 views

Example of a topological space

I have realized that inserting finiteness in topological spaces can lead to some bizarre behavior. For example, it seemed natural to say that every compact subspace of a metric space is closed and ...
5
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1answer
36 views

Hausdorff spaces from filters

I'm sure I'm just being silly, but I've run into a claim in a paper I'm reading which I don't understand. Suppose $\mathcal{F}$ is a filter on $\mathbb{N}$. There is a natural topology $\tau_\mathcal{...
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1answer
23 views

Convergence in topological space

I need to show that a convergent sequence in a metric space $(X,d)$ converges in a topological space $(X,t)$ where $t$ is the topology generated by $d$ on $E$. By definition, for every $\epsilon >...
2
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3answers
31 views

Prove that $\mathcal{T}_1 \subset \mathcal{T}$ with $\mathcal{T}_1$ is the standard topology on $\mathbb{R}$.

Let $\mathcal{B}= \left\lbrace [a,b): a,b \in \mathbb{R}, a<b \right\rbrace$. Let $\mathcal{T}_1$ be the standard topology on $\mathbb{R}$ I have already proved $\mathcal{B}$ is a base of a ...
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3answers
53 views

Is $\mathcal{T}$ a topology on $\mathbb{R}$?

Let $a \in \mathbb{R}$, we have $V_a=(a,+\infty)$, $F_a=[a,+\infty)$. 1.Prove that $\mathcal{T}=\left\lbrace \emptyset,\mathbb{R},V_a:a\in\mathbb{R}\right\rbrace$ is a topology over $\mathbb{R}$. 2....
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2answers
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$[0,1]$ is commonly called the unit interval - is there a similar term for $[-1,1]$?

The interval $[0, 1]$ is commonly called the 'unit interval'. Is there something similar for $[-1, 1]$? Like a pre-defined name.
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6answers
211 views

Closure in a topological space

Let $(X, \tau)$ be a topological space. Let $A,B$ be subsets of $X$. Show that $cl(A \cup B)$ $\subset$ $cl(A)$ $\cup$ $cl(B)$ Proof: Let x be in the closure of $A \cup B$. That means for every open ...
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1answer
27 views

Interior topology

Let $(X,\tau)$ be a topological space. Show that $int(A) \cap int(B)$ $\subset$ $int(A \cap B)$ Proof: x $\in$ $int(A)$ and $x\in int(B)$ means that $\exists$ $U_1 (open)$ containing x so that $U_1 ...
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1answer
15 views

Test if line segment formed by two points intersect with image edges

I have a map that looks roughly like the following picture. The map is discretized into pixels and has size nxn where n is in <...
2
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1answer
28 views

Nagata-Smirnov vs. Urysohn metrization theorems - an example?

I'm looking for an example to demonstrate that fact that the Nagata-Smirnov thm is "more useful" than Urysohn's. That is, I'm looking for a space that you can prove is metrizable using Nagata-Smirnov ...
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0answers
47 views

Given subset $A$ and $ A_x = \{y : (x,y) \in A\} $, Are the following equivalent

(This is Baire-Fubini theorem for categories) For a subset $A \subset \mathbb R^2$ and $x \in \mathbb R$ we denote $$ A_x = \{y : (x,y) \in A\} $$ Are the following equivalent? $A$ is of first ...
4
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1answer
71 views

Fundamental group of $S^3$ with finitely many points removed?

I had a question about the fundamental group if the $3$-sphere with finitely many points removed. Since $\pi_1(\mathbb{S}^{n-1}) \cong \pi_1({\mathbb{R}}^n\setminus\{0\})$, I thought $\pi_1(\mathbb{S}...
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1answer
16 views

Infinite intersection of nested finite open cover of a compact space

In my research, $X$ is a topological space and for every $n\in\mathbb{N}$, $\mathcal{U}_n$ is a finite open cover of $X$ such that for every $n\in\mathbb{N}$, we have $\mathcal{U}_{n+1}\prec \mathcal{...
3
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1answer
37 views

Is a compact metric space with a “doubled” point sequentially compact?

Let $X$ be a compact metric space with topology $\tau$ generated by the metric. Consider a new point $x_1\notin X$ and a non-isolated point $x_0\in X$. Set $\overline{X}=X\cup \{x_1\}$, equipped with ...
2
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1answer
24 views

The Fundamental group of the circle from “Introduction to knot theory”, Ralph H. Fox (3)

In this book,as a third completion (the second completion I have not asked it yet as I am still writing it) to The Fundamental group of the circle from "Introduction to knot theory", Ralph H....
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2answers
65 views

Proving that $(0,1)^3$ not homeomorphic to $[0,3)^3$

What are some of the various ways of proving that $(0,1)^3$ is not homeomorphic to $[0,3)^3$ using the fundamental group and homology groups? I feel like I have various ways of understanding why this ...
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0answers
32 views

Is any space with this property homeomorphic to the three-torus? [on hold]

If you have a three-dimensional, locally Euclidean space such that any path along a coordinate direction is closed (you always eventually come back to your starting point), is this necessarily ...
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1answer
45 views

How do we prove that this set is not manifold? [on hold]

Prove that: $$Z:=\Big[ {(x,y,z) in $R^3$|x^2 +y^2-z^2=0}\Big]$$ is not a manifold.(Even It is not a topology manifold )
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0answers
23 views

Look for a cochain map $ \psi : C_* \to D_* $ [duplicate]

I am looking for a injective cochain map $ \psi : C_* \to D_* $ such as the map {$\psi_i: C_i \to D_i$} is an injective but the map {$\psi_*: H_I(C_*) \to H_i(D_*)$} is not an injective for any $i\geq ...
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2answers
41 views

Topology Hausdorff space

Let $X$ be Hausdorff space and $f$ is a continuous function from $[0,1]$ to $X$. If $f$ is one-one, then image of $f$ is homeomorphic to $[0,1].$ I did something like defining mapping $g$ from image ...
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1answer
18 views

Definition of multi resolution analysis

How can (a) $V_j \subseteq V_{j+1}$ be true, yet (c) is also true? Does that mean that $V_j$ as $j \rightarrow -\infty = \{0\}$?
2
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1answer
41 views

Order topology on $\mathbb{N}$ is discret topology?

Let $\mathbb{N}=\{0,1,2,\dotso\}$ and $(\mathbb{N},<)$ with the usual ordering $<$. Let $\tau_<$ be the order topology with regards to $<$. Then $\tau_<$ is the discrete topology (...
5
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3answers
70 views

Every interval $I \subset \mathbb{R}$ is connected. [Proof clarification]

I struggled to understand a part of the following proof. Topological Proof that every Interval $I \subset \mathbb{R}$ is connected Definition: A topological space is connected if, and only if, it ...
2
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1answer
34 views

Determine $X$ up to homeomorphism.

Let $X$ be a non-empty topological space. Assume that every function $f:X \rightarrow \mathbb{R}$ is continuous. Determine $X$ up to homeomorphism, assuming that $X$ is countable. My try: If $X$ is ...
3
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1answer
28 views

$C^m\to C^{m-1}$ projection map projecting out last factor inducing $G_n(C^{m-1})\to G_n(C^m)$?

Assume $m>n$. Consider projection map $\pi: C^m\to C^{m-1}$ by $(z_1,\dots, z_m)\to (z_1,\dots, z_{m-1})$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$ and endow the topology as a ...
0
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1answer
26 views

Let $h,h'$ be hermitian metric over vector space $V$, then grassmanian $G_n(V_h)\to G_n(V_{h'})$ is always continuous?

Let $V$ be a finite dimensional vector spaces over complex number and choose hermitian metrics $h,h'$ over $V$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$. Since $V$ has 2 metrics, ...
3
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2answers
60 views

Neighbourhoods in topology

Let $X=C[0,1]$ and consider the topology $\tau=\tau(S)$ generated by $$S=\{V_{x,U}\}_{x\in[0,1],~U=(a,b)\subset\Bbb R},$$ where $$V_{x,U}=\{f\in C[0,1]:f(x)\in U\}$$ $1)$ Let $V\in\tau$ be a ...
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1answer
18 views

Prove that the set $\beta$ = {$O(K,U)$ | $K \subset X$ is compact, $U \subset Y$ is open} is a subbasis.

Let $X$,$Y$ be topological spaces and $C(X,Y)$ be the set of continuous maps $X \rightarrow Y$. If $K \subset X$ and $U \subset Y$, define $O(K,U)$ to be the set of $f \in C(X,Y)$ such that $f(K) \...
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1answer
43 views

Every open set is a countable union of compact sets. [In which kind of topological spaces this is true]

Let $X$ be a Hausdorff topological space. Under which hypotheses on $X$ every open subset can be written as a union of countably many compact sets. I was wondering if $\sigma$-locally compact is a ...
2
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0answers
62 views

The Fundamental group of the circle from “Introduction to knot theory”, Ralph H. Fox (1)

The says in the beginning of discussing this title: "Let the field of real numbers be denoted by R and the subring of integers by $J$. We denote the additive subgroup consisting of all integers ...
1
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1answer
51 views

Homotopy equivalence of S1 and R2-0 [duplicate]

I want to show that $X=S^1=\{x^2+y^2=1|x,y\in \mathbb{R}\}$ and $Y=\mathbb{R}^2-\{0\}$ are homotopy equivalent. For this I have to find a function $f:X\rightarrow Y$ and a function $g:Y\rightarrow X$ ...
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2answers
45 views

Which subsets of $\Bbb R$ are a countable union of open sets and countable sets?

From (1), every open subset of $\Bbb R$ is at most a countable union of open intervals. The converse is also true: any countable union of open intervals is an open set. However, I want every subset of ...
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2answers
28 views

Co-Countable topology cases

Let X be the reals. The $\tau$ $=$ $\{$ $\varnothing$ $\}$ $\cup$ $\{$ $A$ $:$ $A^c$ is countable $\}$ What I want to show is that the union of an arbitrary collection of open sets is open. So I ...
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0answers
30 views

Elementary question on Hausdorff distance: Difference between integrals on level sets

Let $U \subset \mathbb R^2$ be compact and consider $h:U \times (0,T) \to \mathbb R$ a uniformly bounded function such that: $0 \lt c_1 \le h \le 1-c_2 $ $f_1,f_2 \in W^{2,1}_p(U \times (0,T))$ for ...
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2answers
97 views

countable family of open sets

Is there a countable family of open subsets of ${\bf R}$ or $[0,1]$ such that each rational belongs to only finitely many of the open sets and each irrational belongs to infinitely many of the sets?
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0answers
51 views

Looking for proof of the Nerve Theorem

Where in literature could I find a nice self-consistent proof of the famous Nerve Theorem? One possible statement is as follows: Let $X$ be a triangulable space and let $\mathcal A = \{A_1,\dots,...
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2answers
37 views

How to create irrational endpoints of open sets in $\mathbb{R}$ from countable basis

I read that $\mathbb{R}$ has a countable basis (i.e. it's second countable). The countable basis consists of open intervals with rational endpoints. Now, from this countable basis, how do you ...
1
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1answer
28 views

Homeomorphic closed copy of the unit ball of the space of bounded linear operators.

Here is an example. Let $X,Y$ be separable Banach spaces and denote by $L(X,Y)$ the Banach space of bounded linear operator $T\colon X\to Y$ with norm $$\| T\|=\sup \{\| Tx\|\colon x\in X, \| x\|\...
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2answers
40 views

Isomorphism, Homeomorphism and the necessity of proving individual invariants

I was wondering if there was a way of avoiding having to prove individual invariants of isomorphism and/or homeomorphism are, in fact, invariants. Consider homeomorphisms. We have to prove that ...
2
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1answer
40 views

Prove that if $X$ is locally compact, then $\epsilon$ is continuous (with the product topology on the domain of $\epsilon$).

Let $X$,$Y$ be topological spaces and $C(X,Y)$ be the set of continuous maps $X \rightarrow Y$. If $K \subset X$ and $U \subset Y$, define $O(K,U)$ to be the set of $f \in C(X,Y)$ such that $f(K) \...
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0answers
30 views

Problem with connectdness and and the diffrential of function.

I was reading lecture notes of Werner Ballmann: Automorphism groups here, and get stuck in the following two Lemma's 2.4 and 2.5, more precisely in lemma 2.4 I did try using flow properties, ie. $\...
2
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2answers
76 views

Showing theres no metric wrt a topology

I have a homework problem that I have no clue how to approach and answer. My topology is kind of weak not entirely sure that I understand the question. My attempt: i) If $V_{1}$ is a sequence of ...
5
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1answer
105 views
+50

Compactness and dimensionality.

Compactness is indeed a central theme in analysis and general topology. Since the first courses in these subjects, we are exposed to criteria for compactness, such as the Heine-Borel Theorem for ...
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3answers
38 views

A topology is closed under finite union and arbitrary intersection of closed sets.

Show that a topology is closed under finite union and arbitrary intersection of closed sets. There are some details that are bothering me in this question. Let $(E,\tau)$ be a topological space. We ...
3
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0answers
56 views

How to prove that $\mathbb{N}$ is closed using any metric function

Let $S$ be a set. I define $a$ as a limit point of $S$ if there a sequence $(a_n) \subset S$ such that $(a_n) \to a$. $S$ is closed if it contains all of its limit points. I know how to prove every ...
2
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3answers
40 views

Any regular space isn't Hausdorff?

A space $X$ is regular if for all $x\in X$ and all closed $F\not\ni x$, there is $U,V$ open s.t. $x\in U$, $F\subset V$ and $U\cap V=\emptyset.$ In wikipedia, they talk about $T_3$ space as Regular ...