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Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

57719 questions
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Why is any compact metric space the union of a countable set a subset which is a perfect space under the induced topology?

Recently I stumbled across the fact that any compact metric space is the union of some countable set and a subset that, when given the induced topology, is a perfect space. Can anyone provide a proof or reference to a proof of this fact? Thanks.
Pierre R.
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Local homeomorphism and coverings

Let $f: X \rightarrow Y$ be a local homeomorphism with X, Y connected, locally path connected, Hausdorff and with X also compact. Then f is also a covering with finite fibers. I know how to show that the fibers are finite. Given that f is a…
Down
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which topology is used on $\mathbb{C}^ 2$ to prove that the complex addition continuous?

The title says it all. The motivation behind this question is that I'm stuck with a theorem in topology, which states that if $f,g:X \rightarrow \mathbb{K}$ (where $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$ - both endowed with the…
temo
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Euclidean space as a sum of non-intersecting and non-parrarel lines

Is it possible to represent $\mathbb{R^{3}}$ as a union of countable non-intersecting and non-coplanar lines?
Konrad Szałwiński
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Proving that a particular restriction of a projection is a quotient map

I was hoping somebody could help me with the following problem: Let $\pi: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be the projection onto the first coordinate, and let $p=\pi|_X$, where $X=(\mathbb{R}_{\geq0} \times \mathbb{R}) \cup…
user121410
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Order topology on the set $X = \{ 1,2 \} \times \mathbb{Z}_{+}$

I am learning topology with the book "Topology" by James R. Munkres and have some difficulty in understanding the Example 4 of section 14 titled "The Order Topology" (page 85 of the second edition). Example 4: The set $X = \{ 1,2 \} \times…
hengxin
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The construction of the Long Line topology

The Long line topology is constructed from the ordinal space $[o,\omega_1)$ ( where $\omega_1$ is the least uncountable ordinal) by placing between each ordinal $\alpha$ and its successor $\alpha + 1$ a copy of the unit interval $I=(0,1)$. But in…
ghb
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Ratio of diameter to area of a set

It may be quite a basic and common thing but I haven't found much after a while of searching and I failed to figure that myself... Let's have a (connected) set $M$ and let $\text{diam}(M)$ be its diameter. How big can his area be? Or in other words…
Jeyekomon
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Continuous function and open set

I'm trying to prove: If $f:A\subset \mathbb{R}^m \to \mathbb{R}^n$ and $A$ is open, then the following statements are equivalent: 1) $f$ is continuous on $A$. 2) $f^{-1}(V)$ is open in $\mathbb{R}^m$, for all $V\subset \mathbb{R}^n$ open. Thanks…
Hiperion
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$A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$

Question is : Suppose $A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$ then $A$ is compact Every subset of $A$ must be compact $A$ must be finite $A$ is disconnected Only compact subsets of…
user87543
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Characterization of closure of set with open neighborhoods

I am trying to prove the following proposition: Let $X$ be a topological space, and $S$ a subset of $X$. then, an element $x$ of $X$ belongs to $\bar{S}$ (the closure of $S$) iff every open neighborhood of $x$ intersects $S$. This is at page two…
Kolmin
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Final-segment and finite-closed topology?

Suppose N is the set of all positive integers and t consists of N, the empty set, and every set { n, n+1, ... } for n any positive integer. This is a topology and is called the “final segment topology.” Is it the finite-closed topology? The…
Zhandele
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If $X$ is not compact, does this mean that Cone($X$) is not compact?

I know that for $X$ a topological space, that if $X$ is compact then so is Cone($X$). Is the following identity also true: If $X$ is non-compact then Cone($X$) is not compact, if not can somebody give a counterexample. Thanks in advance.
user112167
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Prove that the function $f: \mathbb{R}/{\sim} \to S^1$, $f([x]) = (\cos(2\pi x), \sin(2\pi x))$ is a homeomorphism.

Let $\sim$ be the equivalence relation on $\mathbb{R}$ given by $a \sim b$ iff $a-b \in \mathbb{Z}$ Prove that the function $f: \mathbb{R}/{\sim} \to S^1$, $f([x]) = (\cos(2\pi x), \sin(2\pi x))$ is a homeomorphism. $\mathbb{R}$ has standard…
sarah
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Why is the Moore plane not locally compact

I'm trying to show the Moore plane $M$ (or Niemytzki plane) is not locally compact (I'm told it isn't). My guess is problems will arise somehow when considering compact neighbourhoods for a point $(p,0) \in M$, since the basic open sets around…
ZulfiqarIII
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