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; ...

40 views

Topology of $L^2$ space

Cardinality of space of all funcions $f: \mathbb R \rightarrow \mathbb R$ is $\beth_2$. However, cardinality of space of all such square-integrable functions, space $L^2$, is $\beth_1=\mathfrak c$, ...
38 views

Modes of convergence for continuous functions

I just wondered about what modes of convergence for continuous functions $f_n:X\rightarrow Y$ between topological spaces there are. Of course there is pointwise convergence, which is defineable for ...
73 views

34 views

Borel sigma algebra and an extra point

I have a question about Borel sigma algebra on a topological space. Let $E$ be a Hausdorff topological space and $\mathcal{B}(E)$ denotes its Borel sigma algebra. We adjoin an extra point $\Delta$ ...
41 views

Weak /Strong operator topology?

Could someone explain me what weak and strong topologies are and provide some practical example of their use in, for instance, feature checking in data computation and in the study of movement in ...
33 views

Are there known examples of pairs $\left(f, N\right)$, where $f$ is a link invariant that is known to be complete when restricted to link diagrams that have at most $N$ crossings? (Ideally, f should ...
84 views

Converging sequence implies limit point

Is it true that if a sequence in a metric space converges to a value, then that value is a limit point of the set of all terms in the sequence? $E = \{ p_1, p_2, \dots, p_n , \dots \} \subset X$, ...
50 views

Space generated by a reflection

Suppose I embed a mirror (not necessarily plane) in some space (say a manifold). Is there a theory that tells you how to classify the "space" generated by the reflection (the one you see if you were ...
59 views

How to prove a function is harmonic polynomial

1! How to prove this function a harmonic polynomial using Laplace equation For the 1 question I know we can prove harmonic using Laplace Equation but for this on m confused how to start. For the 2 ...
51 views

Is the question in the Munkres's topology book wrong?

At the end of cheapter $8.1$, $4)$ Given spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$. $b)$ Show that if $Y$ is path connected, the set $[I,Y]$ has a ...
270 views

59 views

Importance of metrization theorem?

I wonder if there is a case metrization theorems(such as Nagata-Smirnov, Bing, Urysohn) pave a way to do a theory. What would be a nice application of metrization theorems?
76 views

$E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering

Let $q:E\to X$ be a covering map. Then $E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering. My question is regarding the $"\implies"$ direction: If $E$ is compact, then $X$...
59 views

Showing $\phi(f \cdot g) = \phi(f) + \phi(g)$
For $\phi \in C^1(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$. What I'm not sure is how I'm supposed to relate a ...
Does there exist a continuous function between the following sets: $A.f:(-1,1)\rightarrow (-1,1]$ which is onto and one-one $B.f:\{(x,y):y^2=4x\}\rightarrow \mathbb R$ which is one-one What ...