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
107 views

spectrum of two bounded linear operators

Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.
1
vote
1answer
168 views

Bounded linear operator in weak topology

Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
1
vote
2answers
163 views

Proving something is $1$-Lipschitz

(1) Let $(X,d)$ be a metric space, and let A be a non-empty subset. Show that the function $$D_A :X \to [0,\infty ]$$ defined by $$D_A (x) =\inf \{d(x,y) : y \in A\}$$ is $1$-Lipschitz (when ...
14
votes
5answers
389 views

If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$?

Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ?
1
vote
1answer
93 views

If every point of $S$ is a limit point of $S$ and $S$ is closed, then we say that $S$ is…?

I have a set $S \subset \mathbb{C}$ such that $x \in S \implies x$ is a limit point of $S$ and $S$ is closed Can I conclude that $S$ is connected?
4
votes
0answers
284 views

If $f: X \to Y$, when do we have $\beta Y \supset \overline{f(X)} = \beta X$?

Suppose that $X$ and $Y$ are Tychonoff spaces, denote by $\beta X$ and $\beta Y$ their Stone-Čech compactifications and let $f:X\to Y$ be a continuous map. Using the embedding $Y\hookrightarrow\beta ...
7
votes
1answer
238 views

Clopen subsets of a compact metric space

I am aked to show that in a compact metric space we can find at most countably many subsets which are both: open and close. I would be grateful for your help.
3
votes
2answers
229 views

$f:X\to X$ is one-one and continuous on a compact space. Is $f$ surjective?

Let $(X,\mathcal T)$ be a compact Hausdorff topological space and $f:X\to X$ be one-to-one and continuous. Is $f$ surjective?
2
votes
1answer
200 views

Orthogonal group is a regular submanifold of $GL(n,\Bbb R)$

I want to show that $O(n)$ is a regular submanifold of $GL(n,\Bbb R)$. I think that I can use constant rank theorem but how? I am putting the picture that what I did. Please help me I want to learn. ...
1
vote
2answers
97 views

Is a two-point set a retract of $\mathbb R^3$?

I'm stuck with an example question. Can someone kindly give me some help? Is a two point set a retract of $\mathbb R^3 $ ?
1
vote
1answer
40 views

loops define the same element of a fundamental group?

$X=S^{1}\times I$ and $x_0=(1,0)$ where $1 \in S^1 \subset\mathbb{C} $. Define paths by $f(t)=(e^{2\pi it}, 1), p(t)=(1,t), $and $ h(t)=(e^{2\pi it}, t)$. Do the loops $pf\overline{p}$ and ...
1
vote
2answers
634 views

Continuous functions uniformly convergent to a function, metric spaces, equivalent conditions

Let $X, \ (Y, d)$ be metric spaces, $f_1, f_2, \ldots \ : X \rightarrow Y$ be continuous functions, $f: X \rightarrow Y$ an arbitrary function. Prove that the following condtions are equivalent: 1) ...
6
votes
2answers
119 views

Metric space has at most one isometry other than identity

Could you help me with this problem? Let $d$ be a metric on $[0,1]$ consistent with the standard topology. Prove that the metric space: $([0,1], d)$ has at most one isometry (except for identity). I ...
4
votes
2answers
744 views

Intuition behind the difference between derived sets and closed sets?

I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even ...
2
votes
1answer
45 views

Product of moscow spaces

Let $\{X_a : a\in A\}$ be a family of topological spaces such that $X_K=\prod\limits_{a\in K}X_a$ is a Moscow space of countable $o$-tightness, for every finite subset $K$ of $A$. Then the ...
1
vote
1answer
98 views

Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces?

Let $(X,x)$ and $(Y,y)$ be path-connected pointed topological spaces. Is it true that the statement ''$X$ and $Y$ are homotopy equivalent'' implies ''$(X,x)$ and $(Y,y)$ are homotopy equivalent as ...
1
vote
1answer
49 views

We have $f_n(a_n)=1$ and $(\forall k> n)(f_n(a_{k})=0)$ in a metric space and all $f_n$ are uniformly continuous. Can $(a_n)$ be convergent?

$(a_n)$ is a sequence in the metric space $(X,d)$. For each $n\in \Bbb N$, $f_n:X\to [0,1]$ is a uniformly continuous function and $$f_n(a_n)=1$$ and $$(\forall k> n)(f_n(a_{k})=0)$$ Can $(a_n)$ ...
1
vote
2answers
44 views

Is there a sequence $(x_n)$ with $f_n(x_n)=1$ and $f_n(x_{n+1})=0$ in a compact space?

$(X,\mathcal T)$ is a Hausdorff compact topological space. For each $n\in \Bbb N$, $$f_n:X\to [0,1]$$ is a continuous function and $$x_n\in X$$ is any element such that $$f_n(x_n)=1$$ ...
0
votes
1answer
70 views

Comparability of topologies depends on the basis?

I know that we say that the topology $\mathcal{T}$ on a set $X$ is finer than the topology $\mathcal{T}'$ on $X$ if for any $U\subset\mathcal{T}'$, $U$ is also open in $\mathcal{T}$. And also, ...
1
vote
0answers
37 views

Is the space $X$ in the class dual to the spaces with the Souslin property?

Recall that $X$ is in the class dual to the spaces with the Souslin property: For any neighbourhood assignment $\{O_x: x\in X\}$, there is a subspace $Y \subseteq X$ such that $c(Y)=\omega$ and ...
0
votes
1answer
59 views

A question on arcwised connected spaces

As the tite explains, how to prove that a arcwised connected space is connected space? Thanks for your help.
3
votes
3answers
104 views

How to prove $X$ is normal?

Let $X=[0,\omega_1]\setminus \{\omega_1\}=[0,\omega_1)$. I known every linear order space is hereditarily normal. However, I would like to know the easier proof by which we can conclude that $X$ is ...
0
votes
1answer
157 views

Minkowski functional satisfies triangle inequality on convex sets

New here so apologies if I screw up any decorum. Taking analysis, have a problem set that's tilting toward topology. The problem asks for proof that the Minkowski functional associated with a set K ...
3
votes
2answers
51 views

Question about finite sets/compactness

I understand that every finite subset of a metric space is compact. But are there any topological spaces where finite sets are not compact? Is that even possible? I don't think it is but I just want ...
1
vote
1answer
60 views

On compact topological group

Must a compact topological group be metrizable? If not, is it separable? Thanks for any help.
1
vote
2answers
204 views

Is $L^\infty(\mu)$ a locally compact Hausdorff space?

Here $\mu$ is a probability measure. Another similar question is: is it a $\sigma$-compact space? Thank you in advance!
1
vote
2answers
808 views

Distance between closed and compact sets.

This question is (1-21)(b) from M. Spivak's Calculus on Manifolds. Question: If $A$ is closed, $B$ is compact, and $A \cap B = \emptyset$, prove that there is $d > 0$ such that $||y - x|| \geq d$ ...
2
votes
1answer
107 views

Family of continuous maps generates the topology of X?

I want to know if this is true: Let $X$ be a Tychonoff space and let $\{f_i:i\in I \}$ be a family of continuous maps $f:X\to [0,1]$ such that it separates points and closed sets. Then $\{f_i:i\in I ...
1
vote
1answer
58 views

Extending the function in $\Bbb R^n$

Let's take a function $f:\Bbb R^k \supset A \rightarrow \Bbb R^m$, such that: $(1): \forall_{x,y\in A}:|f(x)-f(y)| \le |x-y|$. Is it true that every such function can be extended to function $f':\Bbb ...
5
votes
2answers
343 views

Question on problem: Equivalence of two metrics $\iff$ same convergent sequences

Community! Im working on the following problem: Let $X$ be a non-epmty set and $d_1,d_2$ metrics on X. Show that the following conditions are equivalent: 1) $d_1$ and $d_2$ are equivalent, i.e. ...
1
vote
0answers
136 views

The Lebesgue number property and uniform continuity (proof check)

Theorem If $f$ is continuous on a compact metric space $X$, then $f$ is uniformly continuous on $X$. Proof Let $\epsilon>0$. For any $y\in X$ there is a $\delta_y$ such that $d(x,y)<\delta_y$ ...
1
vote
2answers
161 views

Is this set closed? (Finding the limit points of a set)

Prove that this set is closed: $$ \left\{ \left( (x, y) \right) : \Re^2 : \sin(x^2 + 4xy) = x + \cos y \right\} \in (\Re^2, d_{\Re^2}) $$ I've missed a few days in class, and have apparently ...
17
votes
10answers
4k views

How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
2
votes
2answers
75 views

Proving the graph of an equivalence relation is closed in a product space

Let $X = \mathbb R^n \setminus \{0\}$, $n\ge2$. Let $\sim$ be the equivalence relation defined by $x\sim y$ iff there exists a non-zero real number $k$ such that $kx=y$. Prove that the graph of ...
3
votes
2answers
345 views

Regular Borel Measures equivalent definition

Please help me understand how the below definition is equivalent to the standard definition of regularity which says "that a measure is regular if for which every measurable set can be approximated ...
3
votes
2answers
90 views

Is every countably compact space feebly compact?

A topological space is said to be feebly compact if every locally finite cover by nonempty open sets is finite. Every compact space is feebly compact but how about countably compact spaces?
5
votes
1answer
232 views

Under what condition only does every compact subset of $X$ is closed implies $X$ Hausdorff?

It is trivial to see that: If $X$ is Hausdorff, then every compact subset of $X$ is closed. I am asking under what condition does the converse hold, i.e. when does If every compact subset of $X$ is ...
1
vote
1answer
64 views

An example for a non-precompact minimal topological group.

Do you have an example of a non-precompact minimal topological group? A topological group $(G,\mathcal T)$ is said to be minimal iff it is Hausdorff and for any compatible Hausdorff topology ...
1
vote
1answer
155 views

There exists a continuous function that satisfies this property

Let $X$ be a non-compact subset of $\mathbb{R}$. I want to show that there a continuous function $f: X \to \mathbb{R}$ such that $f$ is bounded but does not attain its bounds. I think that there ...
3
votes
2answers
43 views

A question on the linear order space

How to see any linear order space (LOTS) is regular? In other words, is it always regular? Thanks for your help. Any help will be appreciated.
6
votes
1answer
2k views

Why is the graph of a continuous function to a Hausdorff space closed?

Say I have two topological spaces given by $(X,\mathscr{T}_X)$ and $(Y,\mathscr{T}_Y)$ where $Y$ is Hausdorff. In addition say I have a function $f:X\rightarrow Y$, and let it be continuous. I want to ...
1
vote
2answers
176 views

Density of $\mathbb{Q}^n$ in $\mathbb{R}^n$

$\mathbb{Q}$ is dense in $\mathbb{R}$ (with the standard topology). I'm pretty sure that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$ too. Is there an easy argument to prove that without reproducing the ...
7
votes
1answer
333 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
4
votes
2answers
50 views

Closedness with respect to an open cover

This is something that came up when I was studying something else, but I am wondering whether the following topological fact is true. Let $X$ be a topological space, and $\{U_i\}_{i=1}^n$ a ...
2
votes
1answer
107 views

Is an ultra-net a subnet of all it's subnets?

Let $(x_d)_{d\in D}$ be a net on a set $X$ and the set $$\{\{x_d\mid d\ge p\}\mid p\in D \}$$ be a base for an ultrafilter on $X$. Let $(x_{d'})_{d'\in D'}$ be a subnet of $(x_d)_{d\in D}$. Is ...
6
votes
2answers
437 views

If the graph $G(f)$ of $f : [a, b] \rightarrow \mathbb{R}$ is path-connected, then $f$ is continuous.

Yesterday I woke up thinking about this question, and I believe I have a proof, but I'm not sure of its validity. Let $\gamma : [a, b] \rightarrow \mathbb{R}^{2}$ be a path from $(a, f(a))$ to $(b, ...
1
vote
2answers
137 views

Necessary and Sufficient Condition for two metrics to have same open sets.

There are couple of independent conditions like one being scalar multiple of another, or if $$d_p(x,y)=(x^p+y^p)^{1/p}$$ then all $d_ps$ and $d_qs.$ which guarantee that open sets are same under these ...
2
votes
1answer
40 views

Analogous notion of knot complements for braids

Knots/links seem to be studied quite a lot for their topological connection to 3-manifolds by considering knot complements in $S^{3}$. Is there an analogous topological entity for braids? They appear ...
7
votes
1answer
495 views

Contractibility of convex set

Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$. Let $Q_m\subset \Omega$ denote ...
2
votes
1answer
50 views

Proving the closedness of given sets

I want to show that the sets $\{(x, y):xy = 1\} $ and $\{(x, y):x^2+y^2 = 1 \} $ are closed in $\mathbb{R}^2$. Geometrically, it is clear that both sets contains all of its limit points in ...