# 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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### In a locally compact Hausdorff space, why are open subsets locally compact?

Let $X$ be a locally compact Hausdorff space, and $A \subset X$ closed. I want to show that $X - A$ is locally compact. I have found a proof here: Open subspaces of locally compact Hausdorff spaces ...
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### The triangle inequality for shortest paths of graphs

In why-the-triangle-inequality I found the statement: for example if $d(a,b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite abstractly,...
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### Extending a continuous map between the boundary of two cells.

I'm working in Lee's book on topological manifolds and have gotten stumped on the first question in chapter 5, the chapter on cell complexes. The problem is: Let $D$ and $D'$ be two closed cells ...
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### “Absolute retracts” in arbitrary category

Is there a standard notion of something like "absolute retract" in arbitrary categories that generalizes absolute retracts in topology? I am mostly interested in categorical approach to Hausdorff ...
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### set of all $2\times 2$ matrcies having neither eigen value is real

Could any one tell me whether the following subsets of $M_2(\mathbb{R})$ are open, closed or neither open nor closed? set of all $2\times 2$ matrcies having neither eigen value is real. set of all ...
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### What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
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### Direct sum of metrizable spaces.

I managed to prove that an arbitrary direct sum of metrizable spaces is again metrizable. However, I used the theorem that says that a hausdorff regular space is metrizable if and only if there existd ...
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### Exposed point of a compact convex set

I'm trying to show that given a compact convex set $K$ in $R^d$, there must be at least one exposed point (where $v$ is exposed if there exists a hyperplane H such that $H \cap K = \{v\}$ . This is a ...
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### 1-1 correspondence between [0,1] and [0,1) [duplicate]

I wonder how to build a 1-1 correspondence between [0,1] and [0,1). My professor offers an example such that 1 in the first set corresponds to 1/2 in the second set, and 1/2 in the first set ...
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### Constructing A Space Filling Curve that fills the Unit Square

I'm reading Neal Carothers' Real Analysis, and he constructs a curve defined over $[0,1]$ that fills the unit square as follows: Let $f$ be a real-valued function over $[0,1]$ such that $f$ is $0$ ...
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### Polish spaces, closed sets and $G_{\delta}$ sets

In a series of lecture notes regarding descriptive set theory, in the section regarding the Borel hierarchy I found the following statement: We will restrict ourselves from now on to Polish spaces,...
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### Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
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### Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
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### Suppose a 2-adic metric is defined. Showing that if $d(x,y)$ has a midpoint, then $x=y$

Let $\mathbb{Z}$ be the integers. Recall 2-adic metric  d(x,y) = \begin{cases} 0 & x=y \\ \frac{1}{2^{n}} & x \ne y\ \text{and}\ 2^{n} \text{is the largest power of 2 that ...
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### Topological features (and / or definition) of homology

I am coming to grips with basic homological algebra as of late, in order to better understand my own subject, that of the study of language. The thing is, I have recently read in some handbook that ...
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### Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
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### Fundamental group of $\mathbb{R}^n\backslash \{0\}$

I am wondering about what the fundamental group of $\mathbb{R}^n \backslash \{0\}$ or more generally $\mathbb{R}^n \backslash U$ where $U$ is a subset of $\mathbb{R}^n$ for $n>1$. For $n=1$ I ...
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### Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...
Please if i have a contractible and closed set $A$ in $X$ thene $A$ is closed and there existe a continuous function $H:[0,1]\times A\rightarrow X$ such that $H(0,u)=u, H(1,u)=p\in X.$ If i ...