Articles of general topology

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: An equilateral triangle can be divided into 3 congruent shapes. Questions: 1) a triangle can be divided into 3 congruent shapes. Is it equilateral? 2) a shape in the plane can be divided into n congruent shapes for any positive integer n. What can it […]

Equivalent form of definition of manifolds.

I am studying topology on my own, and I am having trouble proving the following. For a Hausdorff, connected, locally euclidean paracompact space $X$, there exists a countable basis for $X$. I think if I possibly get any countable open cover of $X$ which consists of coordinate balls (that is, homeomorphic to open ball in […]

Prob. 5, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: Every compact Hausdorff space is a Baire space

This is problem 5 in section 27 of Munkres’ TOPOLOGY, 2nd ed Let $X$ be a compact Hausdorff space; let $\{A_n\}_{n\in \mathbb{N}}$ be a countable collection of closed sets of $X$. If each set $A_n$ has empty interior in $X$, then the union $\bigcup A_n$ has empty interior in $X$. How to show this fact? […]

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle’s center in its interior?

Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ \widehat{G} $ is an abelian group. The topology on $ \widehat{G} $ is generated by the sub-basic […]

Is $[0, 1) \times (0, 1)$ homeomorphic to $(0, 1) × (0, 1)$?

I know how to show $[0,1] \times [0,1]$ is not homeomorphic to $(0,1) \times (0,1)$ by a compactness argument. Is there such an argument that shows $[0,1) \times (0,1)$ is not homeomorphic to $(0,1) \times (0,1)$? If not, what is the best way to show that they’re not homeomorphic?

Semi-partition or pre-partition

For a given space $X$ the partition is usually defined as a collection of sets $E_i$ such that $E_i\cap E_j = \emptyset$ for $j\neq i$ and $X = \bigcup\limits_i E_i$. Does anybody met the name for a collection of sets $F_i$ such that $F_i\cap F_j = \emptyset$ for $j\neq i$ but $X = \overline{\bigcup\limits_i F_i}$ […]

Products of CW-complexes

I am currently reading through May’s “Algebraic Topology” and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using the canonical homeomorphism $(D^{n}, S^{n-1}) \simeq (D^{p} \times D^{q}, D^{p} \times S^{q-1} \cup S^{p-1} \times D^{q})$. However, I remember hearing that a product […]

Domain is Hausdorff if image of covering map is Hausdorff

Suppose that $p:X\rightarrow Y$ is a covering map. Show that if $Y$ is Hausdorff, then so is $X$. I have an answer but I’m not sure if it’s right? By definition of Hausdorff, $\forall x,y, \in Y, x\neq y,\exists U$ open neighbourhood of $x$ and $V$ open neighbourhood of $y$ s.t. $U\cap V =\emptyset$. Let […]

Example of a topological space which is not first-countable

According to Munkres’ Topology: Definition. A space $X$ is said to have a countable basis at $x$ if there is a countable collection $\mathscr B$ of neighborhoods of $x$ such that each neighborhood of $x$ contains at least one of the elements of $\mathscr B$. A space that has a countable basis at each of […]