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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3
votes
1answer
64 views

Predual of $W^{1,\infty}$

I understand the meaning of $u_n$ converges to $u$ weak star (it means that $u_n\in E^*$ and $(u_n,x)_{E^*,E} \to (u,x)_{E^*,E}$ for all $x\in E$) but I've some trouble for identifying a space $E$ ...
0
votes
0answers
19 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
2
votes
3answers
37 views

What's the definition of a “local property”?

Is a property called local if and only if for every point there exists a neighbourhood for which the property is true? For example: Let $X,Y$ be topological spaces. Then $f: X \to Y$ is continuous if ...
1
vote
1answer
20 views

Does this definition of “limit point” really work

I am reading Tapp's introduction to matrix groups for undergraduates. He gives the following definition of limit point of a set: A point $p \in \mathbb R^m$ is called limit point of a subset $S ...
2
votes
0answers
32 views

A more detailed, rigorous proof that a suspension space is not necessarily contractible

Is my answer/proof correct? Please help me make my proof more rigorous and detailed. I need everything to be absolutely clear. Question: Let $X$ be a topological space. The suspension of $X$, ...
-1
votes
0answers
18 views

What is connexity (in simple language)? [on hold]

Please explain the meaning of connexity, connex elements in simple language
2
votes
2answers
43 views

All neighborhoods of a compact subset of an open space are subsets of that open space

Let $K$ be a subset of $U$, with $K$ compact and $U$ open. Prove that there is an $\epsilon > 0$ such that for all $p$ in $K$, a neighborhood of radius $\epsilon$ of $p$ is a subset of $U$. Note: ...
0
votes
1answer
14 views

Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
2
votes
0answers
45 views

Is my Proof Correct and Rigorous: Proving that Quotient Space is Hausdorff

Question: Let $X$ be a topological space and let $A ⊂ X$. Define an equivalence relation $∼$ on $X$ such that the equivalence classes are: • $A$ itself, and, • Singletons {$x$} such that $x /∈ A$. ...
6
votes
0answers
44 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
2
votes
0answers
12 views

Probability Density Function for Randomly Oriented Ellipse

I have an ellipse with a long aspect of a and a short aspect of b. The equation for this ellipse is found on this post: What is the general equation of the ellipse that is not in the origin and ...
1
vote
1answer
48 views

Baire sets in locally compact Hausdorff spaces

(This is a follow-up to Compact $G_\delta$ subsets of locally compact Hausdorff spaces.) Suppose $X$ is a locally compact Hausdorff space. The Baire sets in $X$, denoted by $\mathcal Ba(X)$, comprise ...
0
votes
1answer
32 views

In Hausdorff spaces, compact sets are closed [duplicate]

I've been stuck on this exercise for awhile. It was suggested that I fix a point in the compact set, but I didn't know what to do with that. Can anyone provide an answer?
8
votes
6answers
479 views

What does “removing a point” have to do with homeomorphisms?

I am self-studying topology from Munkres. One exercise asks, in part, to show that the spaces $(0,1)$ and $(0,1]$ are not homeomorphic. An apparent solution is as follows: If you remove a point, ...
1
vote
1answer
102 views

What is an “essential loop”?

I'm a bit confused. Is an essential loop in a topological space $X$ just a loop $\alpha$, which is not-contractible (i.e. $[\alpha] \neq 0$ in the fundamental group of $X$), or is there something more ...
5
votes
2answers
68 views

Is this a criterion for continuity?

Given a topological space $(X,\tau)$ and the product space $(X^2,\tau_2)$. Define the diagonal $\Delta X^2=\{(x,x)\,|\,x\in X\}$ and a set $\mathbf S_\tau=\{\mathcal A\in\tau_2|\Delta ...
1
vote
1answer
28 views

Complement Topology on $S^3$

Given $S^3$ the three dimensional sphere and it's usual Euclidean topology, call it $\tau$, consider $U:=\{(X \setminus A) \mid A \in \tau \}$. Does $U$ form a topology on $S^3$? My guess is no. ...
0
votes
0answers
22 views

Is a topological space locally compact in this particular case? [duplicate]

Sorry if this is a trivial question. If $X$ is a Hausdorff topological space in which every point is contained in a locally compact subset (wrt the topology on $X$), then is $X$ also locally ...
1
vote
0answers
36 views

Is a topological space locally compact in this case? [on hold]

If $X$ is a topological space for which every point is contained in a locally compact subspace then is $X$ also locally compact? Many thanks in advance.
3
votes
1answer
59 views

Extending a uniformly continous function to the closure of its domain

Suppose $X$ is a noraml space and $f:X \rightarrow X$ is continous on X, and also uniformly continous on a subset $A \subseteq X$. In this setting, can one conclude that f is uniformly continous on ...
6
votes
1answer
188 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
1
vote
4answers
57 views

How to exhibit the set of all the limit points of this subset of $\mathbb{R}^k$?

Let $k$ be a positive integer, let $p_0$ be a point in $\mathbb{R}^k$, let $\delta_0$ be a positive real number, and let the set $E$ be defined as follows: $$E \colon= \{ \, p\in\mathbb{R}^k \, ...
4
votes
3answers
63 views

Show $ \{ (\xi,\eta,\zeta) \in \mathbb{R^3} : \xi = \eta = \zeta \}$ is closed

Show $ F =\{ (x_1,x_2,x_3) \in \mathbb{R^3} : x_1 = x_2 = x_3 \}$ is closed. I'd like help finishing of my solution below. Other answers are appreciated as well. It suffices to show that the ...
26
votes
0answers
700 views
+100

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
1
vote
1answer
44 views

If $Y$ is a locally compact topological subspace of $X$ then is $X$ also locally compact?

If I have a topological space $X$ and a subspace $Y$ which is locally compact then does this mean that $X$ is also locally compact? For local compactness, I want to show that every point has a ...
6
votes
1answer
332 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in ...
1
vote
0answers
27 views

Is ${\cal B}=\{ (p_1,q_1)\times (p_2,q_2)\times …| \, p_i-q_i=p_j-q_j \}$ a basis for product topology?

There is a problem from my topology course. For a collection ${\cal B}=\{ (p_1,q_1)\times (p_2,q_2)\times ...| \, p_i-q_i=p_j-q_j \, \forall i,j \in \mathbb{N}\}$. Prove or Disprove: ${\cal B}$ is a ...
6
votes
5answers
333 views

concepts which is present in metric space but not in topological space

I want to know some concepts which is present in metric space but not in topological space. The one that first comes to mind is uniform continuity, equicontinuity i.e. concepts defined with some kind ...
1
vote
1answer
12 views

Continuity of evaluation maps in the topology of compact convergence on $C([0,\infty),\mathbb{R}^{n})$

I'm trying to prove that the evaluation maps $e_{x}:C([0,\infty),\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ given by $e_{x}(f):=f(x)$ are Lipschitz-continuous with respect to the metric ...
2
votes
1answer
27 views

Unit close disc to prove a matrix algebra identity?

I need to prove that every $3 \times 3$ matrix with real positive entries has one eigenvector with a positive eigenvalue. Now, how do I prove this using the fact that the set $B=\{x=(x_1,x_2,x_3)\in ...
2
votes
1answer
21 views

Quick question: Map being smooth vs Graph being submanifold of the product space [on hold]

Is $f:X\rightarrow Y$ smooth if and only if the graph $\Gamma_f$ is a closed submanifold of $X\times Y$? Thank you very much.
1
vote
1answer
28 views

Non-connectedness in the plane

Let us have open $V \subset \mathbb{R}^2$ and $x\in V$. Now, how can I prove that the quotient space $V\backslash\{x\}$ is not simply connected? Pictorially I understand it as the failure of a loop to ...
9
votes
2answers
129 views

Why is the support defined as a closure?

In the definition of the support of a real function $f$ on $X$, why is it important to consider the closure of the set $S=\{x\in X:f(x)\neq0\}$ and not just $S$ itself? Why is the closure of $S$ ...
1
vote
1answer
36 views

$\mathbb{N}$ is a Compact Space with the Co-finite Topology?

Let $X$ be the topological space on the set $\mathbb{N}$ with the cofinite topology. I am having a hard time seeing why this is compact in the topological sense. If each open $n$-hood on $X$ ...
4
votes
1answer
40 views

Quasi Cauchy sequences in general topology?

Suppose $(X,\tau)$ is a topological space and that $(X^2,\tau_2)$ is the product space. Now define $\mathscr S\!_\tau=\{W\in\tau_2|\Delta X^2\subseteq W\}$, where $\Delta X^2=\{(x,x)|x\in X\}$, ...
1
vote
1answer
29 views

Theorem on continuity and closed sets

In the text Mathematical Analysis, Second Edition by Tom Apostol theorem 4.24 states: Let $f:S\rightarrow T $ be function from the metric space $(S,d_{S})$ to another $(T,d_{T})$. Then $f$ is ...
1
vote
1answer
26 views

Determining a finite subcover for a compact topological space

Suppose (X,τ) is a compact topological space and C ⊆ X is closed. Show that C is compact in (X,τ) So far I have come up with: Let P be an open cover of C. Thus C is closed, then its complement, P', ...
2
votes
1answer
62 views

Prove that these loops are homotopic [duplicate]

Let $G$ be a topological group with identity element $e$. Let $f,g: (S^1, (1,0)) \to (G,e)$ be loops in $G$ with base point $e$. We define $f * g: (S^1, (1,0)) \to (G,e)$ by $$f * g(s) = f(t) \cdot ...
5
votes
1answer
68 views

Topology generated by Functions.

Fix a set $X$. Let $\mathcal{F}_0$ be a set of functions $g:X\to\mathbb{R}$. Let $\mathcal{T}_0$ be the smallest topology on $X$ in which all $g\in\mathcal{F}_0$ are continuous. Next, we say that a ...
0
votes
2answers
38 views

Conditions on $A,B$ that are inherited by $A+B$.

Let $A,B$ be subsets of $\Bbb{R}$. Which of the following is false: If $A,B$ are bounded, then so is $A+B$. If $A,B$ are open, then so is $A+B$. If $A,B$ are closed, then so is $A+B$. ...
2
votes
0answers
78 views
+50

Is the inverse limit of simplicial maps between finite directed graphs also a graph?

I think I have an intuitive understanding of why the following statement might be true, but I am not sure how to go about proving it. It's also possible my intuitive understanding is wrong and the ...
1
vote
1answer
98 views

The Kuratowski Monoid

I have been reading the paper "The Kuratowski closure complement theorem" by "B. J. Gardener and M. Jackson". In that the author discusses the 6 different monoid structures as follows: Extremally ...
7
votes
5answers
151 views

Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...
8
votes
1answer
77 views

Topological Hangman

Suppose a mysterious adversary has captured me and challenged me to the following topological game. We fix some finite set, say, $X = \{1, 2, 3, \ldots, n\}$, and my adversary secretly constructs a ...
1
vote
1answer
31 views

If two non-disjoint subsets are connected, why does their union have to be connected?

So X and Y are two sets such that their intersection is nonempty. I want to show that if X and Y are each connected, together their union is connected. I tried proving this by contraposition and I've ...
0
votes
0answers
15 views

A basic doubt on upper semi-continuity of set-valued maps

Upper Semi-Continuity for set valued maps have two definitions $h:\Bbb R^d \to 2^{\Bbb R^d}$ is upper semi-continuous if 1) Sequential definition : $x_n \to x$, $y_n \to y$ and $y_n \in h(x_n)$ ...
2
votes
0answers
34 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
3
votes
0answers
31 views

The weak-star topology is Completely Hausdorff (in particular Hausdorff).

Let $X$ be a normed space, $X^*$ its dual space, $(X^*, w^*)$ is completely Hausdorff. Proof: Let $f, g \in X^*$, $f\neq g$ then $\exists x\in X$ such that $f(x)\neq g(x)$ i.e. $\hat{x}(f)\neq ...
0
votes
3answers
63 views

How to show that a real continous function with image in the rationals is constant?

Can someone please explain to me how I am supposed to approach this question: If $f: [0,1] \to \mathbb{ R}$ is continuous, and has only rational values, then $f$ must be a constant.
1
vote
1answer
45 views

The intersection of infinitely many dense open sets may not be dense

I know that if $A$ and $B$ are dense open subsets of topological space $X$, then $A\cap B$ is also dense open. Furthermore, if $X$ is a Baire space, then the countable intersection of dense open ...