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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.

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How can 1-manifold can be written as a finite union of spaces homeomorphic to [0,1]

I'm studying from Munkres' Topology textbook. There is an example in page 308 that says: Every 1-compact manifold X has topological dimension 1. The space X can be written as a finite union of spaces that are homeomorphic to the unit interval…
user149043
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Uniqueness of the Quotient Topology

Let $q:X\to Y$ be a surjective map, where $(X,\tau_X)$ is a topological space. The quotient topology $\tau_q$ on $Y$ is given as $U\in \tau_q$ iff $q^{-1}(U)\in \tau_X$. Suppose that there is another topology $\tau_d$ on $Y$ such that for any…
SBF
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an example of an interesting connected topological space

Can anybody tell me an example of a connected topological space which every convergent sequence in this space is constant (after a finite number of terms)? thnks!
user115608
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Which is the topology generated by the neighborhood system $V(x)=\{\{x\}\}$

My question is the following: Which is the topology generated by the neighborhood system $V(x)=\{\{x\}\}$ ? I say that is the coarse topology but I don't know how is the mechanism to generate a topology from a basis sorry if this is so trivial but…
LFRC
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what is a usual topology for R

i'm studying topology. Let T be a usual topology for R(real) generated by usual metric. first, i know that elements of T are open sets in R. i wonder form of T(topology). Under given condition, Can T have many form? or is T unique family?
user137002
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Stereographic projection is a homeomorphism $S^n \setminus \{p\} \to \mathbb{R}^n$

Let $S^n$ be the $n$-sphere, $N=(0,0,...,0,1)$ and be the north pole of $S^n$. I am trying to show that stereographic projection gives a homeomorphism $\sigma: S^n \setminus \{N\} \to \mathbb{R}^n$. I know that a map can be constructed by…
Alex Petzke
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Munkres: Compact subsets of Hausdorff Space

Claim:If $A,B$ are compact disjoint subsets of the Hausdorff space $X$, then there exists disjoint open sets $U,V$ containing $A,B$ resp. Would I be on the right track in saying that since $A,B$ are compact subsets of $X$ then choose…
Mr.Fry
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Show $(S^1\times [0,1])/$~ is homeomorphic to $D^2$

Define an equivalence relation on $S^1\times[0,1]$ by: $(x,t)$~$(y,s) \iff xt=st$. Show that $(S^1\times [0,1])/$~ is homeomorphic to the unit disc $D^2$. My attempt: Let $g: S^1\times[0,1]\to D^2$ by $g(x,t)=xt$. Now $g(x,t)=g(y,s)\iff xt=ys\iff…
user124910
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A shrinking map that is not a contraction no fixed point.

Working from Munkres. Show that $f: \mathbb{R}\to \mathbb{R}$ given by $f(x) = [x+\sqrt{(x^2+1)}]/2$ is a shrinking map that is not a contraction that has no fixed point. I figured out the fixed point part but have no clue how to show the other…
EgoKilla
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Metrisability of an Arbitrary Topological space

Is there a general condition to tell whether a topological space is metrisable? (Without having to find the metric explicitly?) Thanks
marc
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Looking for non-trivial topologies satisfying certain conditions

I'm looking for topologies T on an infinite space X which divide the subsets of X into 2 non-empty collections: (1) sets which are both open and closed (clopen); (2) sets which are neither open nor closed. The trivial topology is one example, with X…
MikeC
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Diagonal contained in interior of inverse image of open sets containing the diagonal implies continuity

Let $X, Y$ be topological spaces and let $f:X \rightarrow Y$ be a function and let $g = f \times f : X \times X \rightarrow Y \times Y$. I want to show that if: 1) $Y$ is normal and 2) for all open sets $U$ of $Y \times Y$ which contain the…
user133398
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Prove or disprove the continuity of the Identity function on the topological space?

Let $\tau_1$be the usual topology on $\mathbb R$ . Define another topology $\tau_2$ on $\mathbb R$ by $$\tau_2 = \{U \subseteq \mathbb R \ \ | \ \ U^c \ \ is \ \ either \ \ finite \ \ or \ \ empty \ \ or \ \mathbb R\ \ \}$$ If $I : (\mathbb R ,…
user120386
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Is there a cardinal $\kappa$ for which $\mathbb{R}^\kappa$ is not normal?

I'd like to know if normality is a multiplicative property for $\mathbb{R}$ or there is a cardinal $\kappa$ for which $\mathbb{R}^\kappa$ is not normal.
user34870
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The Long Line is not second countable

Let $\omega_1$ be the first uncountable ordinal. Let $L$ denote $\omega_1 \times [0,1)$ with the order topology and smallest element removed. How can we show this space is not second countable?
user111970
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