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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1answer
48 views

Proof that the Order topology on $\mathbb{R}$ has the same basis as the Euclidean topology

I want to prove that the Order topology on $\mathbb{R}$ has the same basis as as the Euclidean topology on $\mathbb{R}$. Assume that the only thing we know about the order topology is that it has the ...
2
votes
1answer
56 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
0
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1answer
36 views

Show that $A\subseteq B\implies A^{\circ} \subseteq B^{\circ}$ in a different way.

Let $A$ and $B$ be subsets of a metric space $(M,d)$. If $A\subseteq B$, then $A^{\circ} \subseteq B^{\circ}$. Proof : Assume that $a\in A^{\circ}$. Then there exists a $r>0$ such that ...
4
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1answer
58 views

$X$ is locally connected and $f: X \to Y$ is onto where $Y$ has the quotient topology. Prove that $Y$ is locally connected

Those questions about connectivity drive me crazy - I'm having so much difficulty proving them. Say $X$ is a locally connected space, and $f: X \to Y$ is onto where $Y$ has the quotient topology. ...
4
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0answers
48 views

Composition, fibrations?

Let $p: D \to B$ and $q: E \to B$ be fibrations and let $f: D \to E$ be a map such that $q \circ f = p$. Suppose that $f$ is a homotopy equivalence. My question is, does it follow that $f$ is a fiber ...
1
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0answers
35 views

Continuity by composition with a homeomorphism

I only want to know what do you guys think about the following proof. That's an exercise I've tried to do and I don't have an available answer, so... If you find some error or imprecision, I'd be ...
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1answer
32 views

Discrete Topology

For a set X with the discrete topology, show that for every $A\subset X$: $\text{int} A = A$ $\text{ext} A = X\setminus A$ $\partial A = \emptyset$ where int means the interior of $A$, ext means ...
4
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1answer
54 views

Are $\{re^{i\theta}: 0<r\le 1, 0\le \theta < 2\pi \}$ and $\mathbb{R} \times (0,1]$ homeomorphic?

Are $A := \{re^{i\theta}: 0<r\le 1, 0\le \theta < 2\pi \}$ and $B:=\mathbb{R} \times (0,1]$ homeomorphic? My intuition tells me no. And yet, I can not find a single topological property that ...
1
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0answers
89 views
+50

Supportive book(s) for unproven-theorems of General Topology by R Engelking?

I am studying General Topology by R Engelking. And, it has many theorems left without proofs. Some of them are very hard and I don't think the author had intention to leave them as exercises. Would ...
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0answers
19 views

Lemma in Farkas/Kra: Riemann surfaces on construction of domain satisfying certain properties

I'm having some trouble understanding the proof of this lemma. I can follow the construction of $u$ and $D$, however the final step in the proof seems to be without justification. Specifically why ...
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0answers
44 views

Subbasis of Product Topology

There's a thing that confuses me about the product-topology. Namely, we said in class that the sets of the form $U_1 \times \cdots \times U_n \times \prod_{i = n+ 1}^\infty X_i$, where $U_i$ is open ...
3
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1answer
33 views

Say $X$ is $T_2$, $f: X \to Y$ is continuous, $D$ is dense in $X$ and $f|_D :D \to f(D)$ is a homeomorphism. Then $f(D) \cap f(X- D) = \emptyset$

I've been looking into the following question: Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism (It is also given that $X$ is ...
7
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1answer
94 views

$X$ is connected, $A \subset X$ connected, and $C$ a component of $X\backslash A$. Is $\overline A \cap \overline C \ne \emptyset$?

I'm trying to prove or disprove the following statement: If $X$ is connected, $A \subset X$ is connected, and $C$ a component of $X\backslash A$ then $\overline A \cap \overline C \ne \emptyset$. I ...
1
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1answer
46 views

Is every image of a loop in Hausdorff space, homeomorphic to $S^1$?

Let $S^1$ be the 1-sphere with respect to the standard norm on $\mathbb{R}^2$. Let $X$ be a Hausdorff space and $\alpha:[0,1]\rightarrow X$ be a loop. Define $C=\alpha([0,1])$. Define ...
0
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1answer
31 views

Equicontinuous sequence in $C(\mathbb{R^2})$ and Arzela-Ascoli Theorem

Could anyone help with the following problem? I am trying to work out this last practice problem for my Real Analysis prelim but I'm not sure about how to approach it. It looks very similar to the ...
1
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1answer
30 views

$\forall A\subset \mathbb{N}$ the sum of the reciprocals of $A$ diverges iff $A$ is $(\tau, \mathbb{N})$-dense

Show that is possible to endow the natural numbers with a topology $\tau$ such that for every $A\subset \mathbb{N}$ the sum of the reciprocals of $A$ diverges iff $A$ is $(\tau, \mathbb{N})$-dense.
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0answers
19 views

Discrete subspace of a Hausdorff space [duplicate]

Every infinite Hausdorff space has a countably infinite discrete subspace. Now I know $R$ under usual topology has $Z$ and under "lower limit topology" also has $Z$ as such. So goes ...
3
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1answer
61 views

Homeomorphism $\mathbb{R}^{2}\setminus \mathbb Z^2$ to $\mathbb{R}^{2}\setminus \{ (x,y) \ | \ (x-n)^2+(y-m)^2<\frac{1}{10}, n, m \in\mathbb Z \}$

Show that $\mathbb{R}^{2}\setminus \{(x,y)\, |\, x \text{ and } y \text{ integers }\}$ is homeomorphic to the space $\mathbb{R}^{2}\setminus \big\{(x,y) \ | \text{ there are integers } $n$, $m$ ...
7
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0answers
74 views

Is it the case that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?

So as the question statement asks, is it necessarily the case that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism? I suspect the answer is yes, but I don't know ...
2
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2answers
53 views

show that $X$ is homeomorphic to the $n$ dimensional (real) projective space.

Let $D^n=\{(x_1,...,x_n)\in\mathbb{R}^n: \Sigma_{i=1}^{n}x_i^2\leq 1\}$. Let $X=D^n\times\{0\}\cup D^n\times\{1\}$ and let $Y$ be the quotient of $X$ obtained by identifying $(x,0)$ and $(x,1)$ for ...
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0answers
23 views

Order topology on a poset

How does the open rays on a partially ordered set X forms a subbasis for a topology on X (which is called order topology) ? I was considering the case X has only one element. Moreover, I know if the ...
3
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1answer
47 views

$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$

Let's consider a group $M$ (under multiplication) of all matrices $A$ of size $2 \times 2$ over $\mathbb{R}$ so that $\det(A)=1$. How to show that the group is homeomorphic to the $S^{1} \times ...
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0answers
47 views

Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
1
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0answers
36 views

Is the product of two indecomposable continua indecomposable?

We say $X$ is a continuum, if it is a compact, connected metric space. A subset $A\subseteq X$ is said to be a subcontinuum is $A$ is a continuum. A continuum $X$ is said to be decomposable if ...
0
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1answer
36 views

Forming a splitting of $R^2$

In my topology course i recently learned about the concept of a splitting and using it to assist in determining whether a set is connected or not. So I am trying to determine if a subset in $R^2$ is ...
1
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2answers
53 views

Does a proper map have to be continuous?

In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a ...
0
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1answer
40 views

basis for topological spaces…

When we say that the sets $(a,b)$ form a base for usual topology on $\mathbb R$ why do we say in that context that $a,b$ are rational? Why not irrrational? And if we take only open intervals then ...
3
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4answers
107 views

How to show that $S^1$ with one point removed is still connected?

$S^1:= \{x \in \mathbb{R^2}: \|x\| = 1 \}$ Suppose $y_0 \in S^1$. Prove $(S^1-\{y_0\}, \mathcal{T}_{S^1- \{y_0\}})$ is connected. where $\mathcal{T}_{S^1- \{y_0\}}$ is the subspace topology coming ...
1
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1answer
86 views

Example of a homeomorphism on the real line?

I'm to give a short presentation on "basic topology" for a first semester undergrad analysis course. Naturally the professor does not expect me to master the topic, so I'm just trying to get some of ...
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0answers
20 views

What quantities does a local topological region have in 3D?

If we take an infinite solid R3 and cut out a torus and sew it back in with Dehn surgery. This will create a local topological region in R3. I was thinking.. are there any characteristic values ...
0
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1answer
34 views

Mistake in Gabriel-Zisman regarding change-of-base of topological spaces?

In III.2.2 of Gabriel-Zisman, a Proposition is asserted which says that the base of change functor sending $X \to B$ to $X \times_{B} B'$, for any $B' \to B$ commutes with colimits in the $X$ ...
2
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0answers
88 views

Deformation retract of a triangle

Let $X \subset \mathbb{R}^2$ be a triangle equipped with the topology induced by the euclidean topology on $\mathbb{R}^2$ and let $Y \subset X$ be the subset made of two sides of the triangle. I need ...
4
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4answers
362 views

Why do Topologies get “finer”?

Why are topologies with many elements called "fine" and topologies with few elements called "coarse"? It seems as though the finer a topology is, the more likely it is for a function defined from that ...
0
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0answers
48 views

Are Prevarieties irreducible?

In Goertz-Wedhorn, a prevariety is defined to be a connected space with functions that locally is an affine variety (were an affine variety is a space with functions that is isomorphic to the space ...
3
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0answers
44 views

Number of path components of a function space

Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?). What ...
1
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2answers
53 views

One-to-one continuous mapping preserve openness?

I am reading a proof, and I see the following steps. Let $U\subset\mathbb R^n$ be open, and $g:U\to\mathbb R^n$ be one-to-one continuously differentiable where $det g'(x) \neq 0$ for all $x\in U$. ...
1
vote
1answer
38 views

How do I prove that $R^n\setminus R^k$ is homeomorphic to $S^{n-k-1}\times R^{k+1}$?

Let $k,n$ be positive integers such that $k<n$. How do I prove that $\mathbb{R}^n\setminus \mathbb{R}^k$ is homeomorphic to $S^{n-k-1}\times \mathbb{R}^{k+1}$? I tried to put specific integers in ...
2
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0answers
77 views

Conditions for $X\times Y=X\oplus Y$ holds? [closed]

In case $X=\mathbb R$ and $Y=\mathbb R$ the product $X\times Y=\mathbb R^2$ is plane. And, also $X\oplus Y=\mathbb R^2$. In this case $X\times Y=X\oplus Y=\mathbb R^2$. Does it always true? If not ...
4
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2answers
62 views

Do “point of accumulation” and “boundary point” mean the same thing?

In my text it says, if a set $\Omega$ contains all points of accumulation $\{c\}$, then $\Omega$ is closed. I was surprised because people usually use "boundary point" in this context. And further ...
2
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3answers
28 views

The union of open discs $C_n$ in $\mathbb{R}^2$ centered at $(n,0)$ with radius $n$

For each $n\ge 1$, let $C_{n}$ be the open disc in $\mathbb{R}^2$, with centre at the point $(n,0)$ and radius equal to $n$. Then $\mathcal{C} = \cup C_{n}$ is ...
5
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1answer
51 views

Exchange of Limit and Integral with Nets

In topology, we have seen that there are examples of nets so that monotone and dominated convergence do not hold anymore. In particular, we worked with the net $\mathfrak{F}$ containing finite ...
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2answers
42 views

Having difficulties with example 6, pg. 143, in Munkres' Topology.

I don't know how to prove that $A$ has no limit points. I maybe have to prove it by showing that for any point $(x, n), x \not= 1/n$ in $X$, one of its neighborhood does not intersect $A$, but I'm ...
3
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0answers
56 views

Contractible pieces of $GL(n,\mathbb{C})$

Is $GL(n,\mathbb{C})$ contractible for any $n$? My intuition is telling me it is not, because the determinant maps the general linear to $\mathbb{C}\setminus 0$ which is not contractible. If there ...
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0answers
27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
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1answer
37 views

Is the boundary of a compact connected subset of $\mathbb R^n, n>1$ connected? [closed]

Let $A\subset \mathbb R^n$ be compact and connected, where $n\ge2$. Is the boundary $\partial A$ connected?
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0answers
48 views

Why (or when) is the direct limit of compact spaces paracompact?

I'm working through Milnor and Stasheff's Characteristic Classes and got stuck in chapter 5, p.66, where some (supposedly) easy facts about paracompact spaces are assembled. One of these is: ...
2
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0answers
26 views

Topological reflection of pretopological closure operator

Given a pretopological space $(X,\mbox{cl})$ where $\mbox{cl}$ is a pretopological closure operator. How does one find the topological reflection of $(X,\mbox{cl})$? I know of a way namely by ...
2
votes
1answer
73 views

Hierarchy of Mathematical Spaces

I really got lost among all those many different spaces in mathematics, and I got really confused what is special case of what. For example, I knew for long time vector spaces, then Hilbert spaces, ...
0
votes
2answers
78 views

When $X\times (Y\times Z)=(X\times Y)\times Z$ in product topology?

Under what conditions $X, Y, Z$ must have so $X\times (Y\times Z)=(X\times Y)\times Z$? and proof? $X, Y, Z$ are topological spaces and $\times$ represents product topology.
-1
votes
2answers
89 views

Is every Hausdorff space metric? [duplicate]

My question is very simple. I know every metric space is Hausdorff, but the converse is true? anyone knows some counterexample? Thanks