Articles of general topology

If Y is compact, then the projection map of $X \times Y$ is a closed map.

This question already has an answer here: Projection map being a closed map 5 answers

Exponential Law for based spaces

I realize most people work in “convenient categories” where this is not an issue. In most topology books there is a proof of the fact that there is a natural homeomorphism of function spaces (with the compact-open topology): $$F(X\times Y,Z)\cong F(X,F(Y,Z))$$ when $X$ is Hausdorff and $Y$ is locally compact Hausdorff. There is also supposed […]

$\mathbb{R}^{2}$ and $\mathbb{R} \times $ are homotopy equivalent, but not homeomorphic

So, let’s consider $M=\mathbb{R}^{2}$ and $N= \mathbb{R} \times [0, +\infty]$ – two topological spaces. Since $\pi_{1}(M)=\pi_{1}(\mathbb{R}) \times \pi_{1} (\mathbb{R}) = \{0 \}$ (since $\mathbb{R}$ is path-connected). This time, fundamental group of every convex subset of $\mathbb{R}$ is also trivial, so it’s time to conclude that $M$ is homotopy equivalent to $N$. But how to prove […]

A metric space is compact iff it is complete and totally bounded

I’m looking at this theorem from Marsden’s Elementary Classical Analysis, but there’s a part of the proof that I don’t understand. First I’ll state the proof of the direction that if it is complete and totally bounded then it is compact. The bold parts of the proof are what I don’t understand. Proof: Assume that […]

When does a dense subspace destine the weak topology?

Let $E$ be a locally convex space, let $E^{\prime}$ be its continuous dual space and let $F$ be a subspace of $E^{\prime}$ which is dense with respect to the strong topology on $E^{\prime}$ (i.e. the topology of uniform convergence on bounded subsets on $E$). Under what assumptions on $E$ is it possible to conclude that […]

Weak topology and the topology of pointwise convergence

I’m reading the definition of weak topology in Banach Algebra Techniques in Operator Theory by Douglas: According to an article about the product topology in Wikipedia, the product topology is also called topology of pointwise convergence. I’m confused with the underscored sentence. According to the answer by @Brian M. Scott to the question What is […]

Topology of pointwise convergence

I’m trying to understand the topology of pointwise convergence, we’re defined it on the set $\cal{F}(X)$ of real functions on set $X$ to be the sub basis with topology $$\{f \in \cal{F}(X) : a < f(x) <b\} $$ where $x \in X$ and $a,b \in \mathbb{R}$. Then it says: ‘A set from the sub-basis consists […]

Closure of $A \subset \mathbb{R}$

The closure of $A$ can be equal to $\operatorname{int}(A)\displaystyle\cup\operatorname{bdry}(A)$. Another definition is that the closure is the set of limit points of $A$. How are these 2 definitions equivalent?

Show $f$ is a surjective isometry.

Give $f:(\mathbb{R}^2,d^{1}) \rightarrow (\mathbb{R}^2,d^{\infty})$ let $f$ be defined $$f(x_1,x_2)=(x_1+x_2,x_1-x_2)$$ My ideal is to try to show that $f$ preserves the origin and that $f$ is continuous and injective, thereby implying $f$ is surjective. I wonder if it is possible to show that $$d^{1}(x,y) = d^{\infty}(f(x),f(y))$$ to show that $f$ is an isometry, but I do […]

Covering map is proper $\iff$ it is finite-sheeted

Prove that a Covering map is proper if and only if it is finite-sheeted. First suppose the covering map $q:E\to X$ is proper, i.e. the preimage of any compact subset of $X$ is again compact. Let $y\in X$ be any point, and let $V$ be an evenly covered nbhd of $y$. Then since $q$ is […]