# Tagged Questions

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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### The $n$-disk $D^n$ quotiented by its boundary $S^{n-1}$ gives $S^n$

Define $D^n = \{ x \in \mathbb{R}^n : |x| \leq 1 \}$. By identifying all the points of $S^{n-1}$ we get a topological space which is intuitively homeomorphic to $S^n$. If $n = 2$, this can be ...
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### When is the union of topologies a topology?

The union of two topologies on some set may or may not be a topology. When is it a topology?
Let $X$ be a $T_1$ space. Let $\mathfrak {D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. Show that there is at most one point belonging to $\... 0answers 169 views ### Category of metric spaces versus category of non-empty spaces Denote by$\mathbf{Met}$the category of metric spaces with metric maps as morphisms. A function$(X,d)\xrightarrow{\ f\ }(X',d')$is called metric if for every pair of points$x,y\in Xwe have d(x,... 1answer 97 views ### Surjectivity of a continuous map between \mathbb{R}^ds Let f:\mathbb{R}^{d}\to\mathbb{R}^d be a continuous map. Show that if \displaystyle\sup_{x\in\mathbb{R}^d}|f(x)-x|<\infty, then f is surjective. I encountered this problem more than 3 ... 3answers 1k views ### What is the interior of a singleton? I was wondering what can be said about the interior of \{{4}\}, the empty set? The interior of a set A is the largest open set contained by A. Hence, if the set at hand is a singleton, then isn'... 2answers 356 views ### Proof of (0,1) is not compact with usual metric. In the proof we say \left\{\left(\frac1n,1\right):n\geq 1\right\} is an infinite cover with no finite subcover. But, (0,1) set also belongs to cover mentioned above. We can say \{(0,1)\} is a ... 2answers 588 views ### Spaces with the property: Uniformly continuous equals continuous I found a nice book about functional analysis with a nice theorem in it: Continuity at 0 is equal to Lipschitz continuous for linear maps in normed spaces. This fact inspires me to ask: Are there ... 2answers 407 views ### Second Countability of Euclidean Spaces Sorry I know this is a stupid question. However I got stuck on this for quite a while. I'm trying to prove that Euclidean spaces have a countable base, which can be constructed by taking all the open ... 1answer 2k views ### Orthogonal matrices forms a compact set [duplicate] Could someone help me to prove that the set of all n\times n orthogonal matrices is a compact subset of \mathbb{R}^{n^2}. I don't know how it can be done. Thanks. 3answers 1k views ### [0,1]^{\mathbb{N}} with respect to the box topology is not compact could anyone help to show that [0,1]^{\mathbb{N}} with respect to the box topology is not compact? Thank you! 1answer 162 views ### Every open subset of \mathbb{R} can be expressed uniquely as a disjoint union of open intervals. Does this generalize to \mathbb{R}^n? I know that every open subset of \mathbb{R} can be expressed uniquely as a disjoint union of open intervals. Further, only countably many intervals feature in any such decomposition. Supposing we ... 2answers 2k views ### Measure on topological spaces So, given a topological space S we can construct its Borel sigma-algebra \mathcal{B}(S). Does it mean that we can construct a measure \mu on this sigma-algebra as well? Say, discrete topology on ... 1answer 293 views ### Tychonoff theorem (1/2) I was trying to prove Tychonoff theorem. First I used (which I showed also): The following are equivalent (a) X is compact (b) every filter of closed set F on X has non-empty ... 2answers 292 views ### Compact metric space group Iso(X,d) is also compact Could you tell me how to prove that if metric space (X,d) is compact, then the group Iso(X,d) is also compact? The group Iso(X,d) is considered with topology determined by a metric \rho on ... 3answers 798 views ### Prove that \Bbb R^2 - \{0\} is homeomorphic to S^1 \times \Bbb R. No idea where to even begin. There is a hint: this requires construction of an explicit function. 1answer 561 views ### Is the product of Polish spaces Polish? If T_t=(E_t,\tau_t) are Polish, t\in I, is T^I:=(\prod_{t\in I}E_t,\prod_{t\in I}\tau_t) Polish? (\prod_{t\in I}\tau_t being the product topology.) If this is not the case, what extra ... 1answer 629 views ### Finite unions and intersections F_\sigma and G_\delta sets Why is the intersection of finitely many F_\sigma sets an F_\sigma set, and the union of finitely many G_\delta sets a G_\delta set? 2answers 82 views ### Continuous mapping from open set to open set Suppose we have two open and bounded sets, \Omega_1,\Omega_2 \in \mathbb{R}^2. Is there a continuous function \textbf{f} mapping \Omega_1 onto \Omega_2? \begin{align*} \Omega_1 & = \{(x,... 3answers 287 views ### How to prove the group of automorphisms of S^1 as a topological group is \mathbb Z_2? The title basically says it all. How does one prove the group of automorphisms of S^1 (the unit circle in \mathbb C), as a topological group, is \mathbb Z_2? I was surprised not to find the ... 2answers 84 views ### Lie group step in proof Let X_e,Y_e \in T_eG be vectors and G = GL(n). Then the right translation is given by Y_g = Y_e g and X_g = X_e g. Now, I have a proof showing that [X_e,Y_e] \in T_eG is the element ... 1answer 1k views ### Proof that Sorgenfrey plane is not normal using points x × (-x) I'm making Exercise 9 of paragraph 31 in Munkres, which is a proof that the Sorgenfrey Plane \mathbb{R}_l^2 is not normal. I'm having trouble on part c of the question. The full question is: Let A... 2answers 534 views ### Open Dense Subset of M_n(\mathbb{R}) Well, I know the fact that GL_n(\mathbb{R}) is open set in M_n(\mathbb{R}), how to show that it is dense also? Well I thought like this: If A\in M_n(\mathbb{R}) and If \lambda_1,\dots,\lambda_n... 1answer 724 views ### Continuous image of the intersection of decreasing sets in a compact space Suppose B_{\epsilon} are closed subsets of a compact space and B_{\epsilon} \supset B_{\epsilon'} \quad \forall \epsilon > \epsilon'. Furthermore, B_0 = \bigcap_{\epsilon>0} B_{\epsilon}. ... 2answers 95 views ### Why Z_p is closed. Let A_n=\mathbb{Z}/p^n\mathbb{Z} be a ring and p is prime, \phi_n: A_n\rightarrow A_{n-1} be a natural homomorphism (Elements of A_{n} define in an obvious way elements of A_{n-1}). Define ... 1answer 1k views ### boundary of the boundary of a set is empty I am learning some stuff about the interior, closure and boundary of sets A\subset\mathbb R^n and I am wondering about the following: 1) \partial\partial A=\partial A ? 2) \partial\partial\... 2answers 1k views ### Extension of a Uniformly Continuous Function between Metric Spaces Let (X,d) and (Y,d') be metric spaces with (Y,d') complete. Let A\subseteq X. I need to show that if f:A\to Y is uniformly continuous, then f can be uniquely extended to \bar{A} ... 2answers 1k views ### What does order topology over Ordinal numbers look like, and how does it work? Space of ordinal numbers are one of the favorite examples of my professor in general topology. I quite understand the idea at the base of ordinal numbers (few things or nothing about the concept of ... 1answer 230 views ### Is there a topological space which is star compact but not star countable? A topological space X is said to be star compact if whenever \mathscr{U} is an open cover of X, there is a compact subspace K of X such that X = \operatorname{St}(K,\mathscr{U}). A ... 2answers 66 views ### Examples of a quotient map not closed and quotient space not Hausdorff Is there any example of a closed relation \sim on a Hausdorff space X such that X/\sim is not Hausdorff? Also, is there any example of a closed relation ~ on a Hausdorff space X such that a ... 2answers 405 views ### A sort of inverse question in topology Given topological spaces X and Y, we often consider the collection of continuous functions, f: X \rightarrow Y. My question is, given two sets X and Y, and a sub-collection \{g_{i}\} of ... 5answers 424 views ### Compactness in \mathbb{Q} Proving that [0,1]\subset\mathbb{R} is compact you make use of completeness and this is a fundamental step in order to characterize compact subsets of \mathbb{R}. Trying to state an analogous ... 1answer 134 views ### Exists homeomorphism which carries each fiber isomorphically to itself, composition? Let \mu and \mu' be two different Euclidean metrics on the same vector bundle \xi. How do I see that there exists a homeomorphism f: E(\xi) \to E(\xi) which carries each fiber isomorphically ... 9answers 1k views ### Why did we define the concept of continuity originally, and why it is defined the way it is? The concept of continuity is a very important idea in topology. Though I am using it all the time, but indeed I don't know what is the original purpose for us to define this concept. And I also don't ... 1answer 324 views ### \mathbb A^n(k) and \mathbb A^n(k)\setminus \{0\} are not homeomorphic Let k be an algebraic closed field. Why \mathbb A^n(k) and \mathbb A^n(k)\setminus\{0\} (for n>1) are not homeomorphic with respect to the Zariski topology? 4answers 8k views ### Cauchy sequence is convergent iff it has a convergent subsequence Prove that if \left ( x_{n} \right ) is a Cauchy sequence in a metric space X then \left ( x_{n} \right ) is convergent if and only if \left ( x_{n} \right ) has a convergent subsequence. Note: ... 1answer 318 views ### On different definitions of neighbourhood. I am going through the basics of topology, mainly to refresh them. I had taken a course some years ago but never used topology actively. So I am reading Munkres's Topology. I have noticed that he ... 2answers 318 views ### Is every suborder of \mathbb{R} homeomorphic to some subspace of \mathbb{R}? Let X \subset \mathbb{R} and give X its order topology. (When) is it true that X is homeomorphic to some subspace of Y \subset \mathbb{R}? Example: Let X = [0,1) \cup \{2\} \cup (3,4], ... 2answers 412 views ### A set which is neither meagre nor comeagre in any interval. As the title suggests, I'm interested in a subset of the real line which is neither meagre nor comeagre in any interval. Does anyone have an example? Added. See the comments for some discussion ... 1answer 289 views ### Every path has a simple “subpath” I've been thinking about this for a while, and can't seem to find any way to do it despite the statement itself seeming obvious. The problem is: Let f:[0,1] \to \mathbb{R}^n be a continuous map,... 2answers 291 views ### Do all manifolds have a densely defined chart? Let M be a smooth connected manifold. Is it always possible to find a connected dense open subset U of M which is diffeomorphic to an open subset of R^n? If we don't require U to be ... 3answers 780 views ### variant on Sierpinski carpet: rescue the tablecloth! I was playing around with Sierpinski carpets (see pretty GPU-produced picture here), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell ... 1answer 287 views ### Product of spaces is a manifold with boundary. What can be said about the spaces themselves? Suppose I have two topological spaces X,Y and I know that X\times Y is homeomorphic to a manifold with boundary. Can I conclude that X and Y are manifolds (maybe with boundary)? If not, ... 3answers 744 views ### What's wrong with this definition of continuity? Consider this definitions: A function f:X \to Y is continuous at x\in X iff for any open neighborhood V_{f(x)} of f(x) there is an open neighborhood U_{x} of x that gets mapped by f ... 3answers 6k views ### Union of closure of sets is the closure of the union: true for finite, false for infinite unions Let A_i be a subset of a metric space for each i\in \mathbb{N}. Let B_n := \bigcup_{i=1}^n A_i. Prove (for any) n \in \mathbb{N} that \overline{B_n} = \bigcup_{i=1}^n \overline{A_i}. ... 1answer 295 views ### In the Sorgenfrey plane, is an open disc homeomorphic to an open square? In the sorgenfrey plane \mathbb{R}_l^2, the subspaceX=\{(x,y):x^2+y^2\leq 1\}$$is not homeomorphic to the subspace$$Y=\{(x,y):|x|\leq 1,|y|\leq 1\},$$because there is only one isolated point ... 1answer 141 views ### Does a map between topologies determine a map between sets? Let (X,\mathcal{A}) and (Y,\mathcal{B}) be Hausdorff spaces. Consider a function \begin{equation*} \phi:\mathcal{B}\rightarrow \mathcal{A} \end{equation*} which preserves inclusion, arbitrary ... 3answers 2k views ### Is a discrete set inside a compact space necessarily finite? Is it true that if A is discrete as a subspace of X, and X \; is compact, then A is finite? If this doesn't hold, then does it hold for X\; manifold? 2answers 372 views ### No Smooth Onto Map from Circle to Torus My professor was lecturing today and he made this statement which I was unable to verify. (I worded it nicer) There is no map which is both smooth and onto from S^1 to S^1$$\times $S^1$. When ...
Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...