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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Example of a homeomorphic map $T:X→Y$

Definition. Let $X$,$Y$ be metric spaces.Then a map $T:X\to Y$ is an homeomorphism if $T$ is continuous, open and bijective. I don't find a counterexample of such maps, may someone give me at least ...
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220 views

Continuous image of a Paracompact space need not be Paracompact.

The following is an exercise in Topology by Munkres. Show that if f is a continuous map from X to Y where X is paracompact then the subspace f(X) of Y need not be paracompact. I am having trouble ...
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156 views

Refinements of open covers in paracompact spaces

For lack of a better term and for this purpose only, let us call a refinement $\mathcal{V}$ of a cover $\mathcal{U}$ faithful if $\mathcal{V}$ can be written as $\mathcal{V}=\{V_U:U\in\mathcal{U}\}$ ...
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134 views

Elaborate on $A^{c}:=\{p\in\mathbb Q : 0<p<\sqrt{2}\}$ not open and not closed in $\mathbb R$

I know that $\sqrt{2}\not\in\mathbb Q$ and $\sqrt{2}\in\mathbb R$ but it is not obvious to me why $\{p\in\mathbb Q : 0<p<\sqrt{2}\} \subset \mathbb R$ is not open. If it is not open, it means ...
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259 views

Pinched torus generalization

The pinched torus is homeomorphic to a sphere with two (different) points identified.           What is the name and topological structure of the ...
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90 views

looking for an imbedding of the Torus in 3-dimensional euclidean space

does anyone know an explicit imbedding $h\colon T^2 \to \mathbb{R}^3$ of the torus $T^2=\mathbb{S}^1 \times \mathbb{S}^1$ into $\mathbb{R}^3$ ? Thanks in advance !!! Cheers...
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36 views

Continuous Extension of $S^{n-1} \to [0,b]$ to $B^n \to [0,b]$.

Let $f : S^{n-1} \to [0,b] \subset \mathbb{R}$ be a continuous function. Does there exist a continuous extension $F : B^n \to [0,b]$ of $f$ that is strictly positive on $\mathrm{Int} (B^n)$?
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361 views

Is this set connected in the Zariski topology?

If T is the set $\mathbb{R}$ with the zariski topology then is the set $X=\{0,1\}$ connected? I think it is connected because the only nonempty subsets are {0,1}, {0}, {1} which are all closed under ...
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185 views

Real Projective Plane copies

how do I show that there is homeomorphism $f:H^2\longrightarrow S^2$, where $H^2$ is the closed upper hemisphere with antipodal equator points identified and $S^2$ is the sphere with antipodal points ...
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97 views

continuous $f:X\rightarrow X'$, $X,X'$ metric spaces properties

True or False, prove or provide a counterexample. (a) If $B\subseteq X'$ closed, then $f^{-1}(B)\subseteq X$ is closed. (b) If $B$ is a bounded subset of $X'$, then $f^{-1}(B)$ is ...
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210 views

Why is $\operatorname{cl}(A)$ the smallest closed set containing $A$?

If $\operatorname{cl}(A)$ is the intersection of all closed subsets of $X$ containing $A$. How to prove that $\operatorname{cl}(A)$ is the smallest closed set $C$ in $X$ such that $A ⊆ C$? I have ...
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581 views

First-countable spaces, closure of a subset and limit of a function

From Wikipedia One of the most important properties of first-countable spaces is that given a subset $A$, a point $x$ lies in the closure of $A$ if and only if there exists a sequence ...
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697 views

accumulation points

Does the set A= {$1+\frac{(-1)^n}{n}:0\leq n\leq10,n\in\mathbb{N}$} have any accumulation points in $\mathbb{R}$? My guess is no since this is a finite set. So there exist $\epsilon$ : $\forall a\in ...
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394 views

Closure in topology proof?

I want to prove $A$ is closed iff $\overline{A}=A$; I need to use the definition of neighbourhoods instead open sets and not use the complement to prove this. So wondering how can you prove it just ...
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654 views

Openness and Closedness in Metric Spaces

Let $X$ be a metric space. Furthermore, let $E$ be an open subset of $X$. Then, the complement of $E$, or all members of $X$ that are not in $E$, is closed, or contains all of its limit points. I ...
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232 views

Is this function continuous?

Let $Y=\oplus_{i \in N}Y_i$ and $Y_i$ is the metriable space $[0,1]$ for each $i \in N$. $x_n \in Y$ denotes that $x_n \in Y_n$ and $0\le x_n \le 1$. $d$ is the usual metric on $[0,1]$. We define a ...
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433 views

Limit of sequence of sets - Some paradoxical facts

I am particularly confused with alternative formulas describing the inner and outer limits of a sequence of sets in topological spaces. The inner limit of a sequence of sets ...
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119 views

Limit points of $ \{x_n, n\in N\}$ as a subset and limits of all its subsequences

In a topological space, let $A$ be the set of limit points of $ \{x_n, n\in N\}$ as a subset, and let $B$ be the set of limits of all subsequences of $ \{x_n, n\in N\}$ as a sequence. Is $A = B$? My ...
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216 views

If a topological space has $\aleph_1$-calibre and cardinality at most $2^{\aleph_0}$ must it be star-countable?

If a topological space $X$ has $\aleph_1$-calibre and the cardinality of $X$ is $\le 2^{\aleph_0}$, then it must be star countable? A topological space $X$ is said to be star-countable if whenever ...
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786 views

Compact but not sequentially compact question

At this page: http://planetmath.org/encyclopedia/SequentiallyCompact.html you can find an example of a compact but not sequentially compact space. My question is: how to prove the existence of "$r ...
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266 views

Is nonempty intersection with every set in a base a sufficient condition to prove the closure of the set is the whole space?

I've noticed in some proofs, if $A\subset B$ for $B$ a topological space, it seems that it is enough to show that $A$ intersects every set in a base of $B$ to prove that the closure $\overline{A}=B$. ...
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265 views

Completely regular space and existence of a continuous function

This is problem $1$, Section $3$, page $252$ of Dugundji's book. Let $X$ be a completely regular space, $C \subset X$ compact and $U$ an open set containing $C$. Prove there exists a continuous map ...
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136 views

Find open sets in a normal space

Let $X$ be a normal space and let $U_{1}$, $U_{2}$ be open subsets of $X$ such that $X= U_{1} \cup U_{2}$. Show that there are open sets $V_{1}$ and $V_{2}$ such that $\overline{V_{1}} \subset U_{1}$, ...
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50 views

A space is $T_{0}$ if and only if the closures of singletons are distinct

This is exercise $1.5$A from Engelking's book, page $47$. Verify that $X$ is $T_{0}$ if and only if $\overline{\{x\}} \neq \overline{\{y\}}$ for every pair of distinct points $x,y$. My try: Suppose ...
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371 views

Why does a constructible set in a Noetherian topological space contain an open subset dense in its closure?

In a Noetherian topological space, a constructible set is a finite union of locally closed sets. This is a conclusion on constructible sets: Every constructible set contains a dense open subset of ...
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266 views

injectivity of a group action

The action of a group $G$ on $X$ is always "injective" in the following sense: if $x\not = y$ then $\forall g\in G$, $gx\not = gy$ indeed if $gx=gy$ then ...
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84 views

quotient of a hyperplane by the action of cyclic group

let $H=\{(x,y,-x-y)\in \mathbb C^3\}$ and let $S^3$ the unit sphere in $H$. Why the following is true : The linear action of $\mathbb Z_3$ on $S^3$ is free and $H/\mathbb Z_3=C(M)$ the cone on ...
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274 views

Alternative Construction of Lens Space

Let $p,q$ be two relatively prime positive integers.In euclidean 3-space we take a regular p-gonal region $P$ with centre of gravity origin.$a_{0},a_{1},....,a_{p-1}$ are the vertices of the regular ...
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156 views

If a connected open set is evenly covered, then its preimage is uniquely partitioned into slices

This is from Topology by Munkres: Let $p:E \to B$ be a covering map. Suppose $U$ is a open set of $B$ that is evenly covered by $p$. Show that if $U$ is connected, then the partition of ...
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710 views

How to find homology groups of a connected sum?

I need to find homology groups for the following simplicial complex: $RP^2$ # $RP^2$ # $\Delta$ # $\Delta$ How to do it? If I am not mistaken, $RP^2$ is not orientable - so we cannot just sum the ...
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208 views

Homotopy equivalence of Unit Balls [duplicate]

Possible Duplicate: $S^n \backslash S^m $ homotopy equivalent to $ S^{n-m-1} $ I'm trying to show that, if we embed $S^m$ in $S^n$ as the subspace $\{ (x_1,x_2, \ldots, x_{m+1}, 0, 0, ...
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135 views

Is N the cts image of the Sorgenfrey line?

I have this question: Prove/disprove: The set of natural numbers (including zero) with usual topology is the continuous image of the Sorgenfrey line. Can't we take the map $g: \mathbb{R}_{l} ...
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1answer
207 views

When does a subbase of a base generate the same topology?

Suppose that $\mathcal{B}$ is a base for a topology on a space $X$. Is there a nice way of thinking about how we can modify $\mathcal{B}$ (for instance, to simplify computations) without changing the ...
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145 views

$F_{\sigma}$ subsets of $\mathbb{R}$

Suppose $C \subset \mathbb{R}$ is of type $F_{\sigma}$. That is $C$ can be written as the union of $F_{n}$'s where each $F_{n}$'s are closed. Then can we prove that each point of $C$ is a point of ...
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22 views

Product (arbitrary) of open functions is open.

Let $f_{\alpha}\colon X_{\alpha}\to Y_{\alpha}$ be open, for all $\alpha \in J$. Then $\prod_{\alpha} f_{\alpha}\colon \prod_{\alpha}X_{\alpha} \to \prod_{\alpha}Y_{\alpha}$ is open? Both $ ...
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38 views

Box topology and axiom of choice

Below is the definition of box topology: Given an indexed family of topological spaces $X_\alpha $, the collection of all sets of the form $$\prod_{\alpha\in J} U_\alpha,$$ where $U_\alpha$ is open ...
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1answer
43 views

simplicial homology definition

I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105: Hatcher ...
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41 views

Show that S is closed but not compact

Show that $S$={$(x,y,z)\in \mathbb R^3: x^3+y^4-z^2=1$} is closed but not compact where $\mathbb R^3$ is the usual topology. Can anyone explain how to go about answering this? I have to show that ...
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27 views

A Hausdorff, Baire space must be σ -compact?

Must a Hausdorff Baire space be $σ-$compact? A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. A ...
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31 views

Closeness of infinite union of closed sets

Is the set $\bigcup_{x \geq 0} \left\{\frac{1}{x+1} \right\} $ closed? For all $x \geq 0$, the set $\left\{\frac{1}{x+1}\right\}$ is a single point, therefore it is closed. But I am not sure about ...
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27 views

it is possible to define a topology on particular vector space

If i take $V$ a finte dimensional vector space on the real number (or complex number). Setting $n=dim_{\mathbb{R}}(V)$, i know that there is a isomorphism of vector spaces so $V \simeq \mathbb{R}^n$. ...
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40 views

Continuity of a product of two real valued continuous function.

In the basic analysis, we proved the following with the $\epsilon - \delta$ method. For metric spaces $X$ and $\mathbb{R}$, $fg:X\rightarrow \mathbb{R}$ is continuous, if $f,g:X\rightarrow ...
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53 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
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36 views

Is this bullet really needed in Furstenberg's proof of infinitude of primes?

See here . The bullet I'm referring to is: Any union of open sets is open: for any collection of open sets $U_i$ and $x$ in their union $U$, any of the numbers $a_i$ for which $S(a_i, x) \subset ...
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17 views

Topology induced by seminorms and initial topology

Let's say we have a family of seminorms $(\rho_\alpha)_{\alpha \in A}$ on a vector space $V$. There are two ways to topologize $V$ using those seminorms: We define topology $\mathcal S$ by a ...
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34 views

Let $A = [0,1) \cup (3,4]$ be a subset of $(\mathbb R, \mathfrak T_H)$

Let $A = [0,1) \cup (3,4]$ be a subset of $(\mathbb R, \mathfrak T_H)$ $\mathfrak T_H$ is the collection of all subsets of $U$ of $\mathbb R$ such that either $U = \emptyset$ or for each $x \in U$ ...
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62 views

Is it always true that the complement of a closed set is open?

In a metric space $M$, by definition, a subset $F$ is closed if $M\setminus F$ is open. However in a general topological space $T$, say with topology $\mathcal{T}$ is this always true? For example ...
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31 views

Let $(X, \mathfrak T)$ be a topological space and let $A$ but a subset of $X$ then $Int(Bd(A)) = \emptyset$

Let $(X, \mathfrak T)$ be a topological space and let $A$ but a subset of $X$ then $Int(Bd(A)) = \emptyset$ I need to decide if this is true or not. I have done a little research and some ...
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45 views

Sum of open/closed/compact sets in $\mathbb{R}^n$ open/closed/compact

I know that the following exercise you can find on internet maybe with solution too, but I want to know, if my "solutions" are correct. Let $X,Y\subset \mathbb{R}^n$, $X+Y=\{x+y;x\in X, y\in Y\}$. ...
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54 views

Example 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be open?

Let $N$ be the following subset of $\mathbb{R}^2$: $$N \colon= \{ \ (x,y) \in \mathbb{R}^2 \ \colon \ \vert x \vert < \frac{1}{y^2+1} \ \}.$$ Then intuitively it is apparent that $N$ is open. ...