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Results tagged with gn.general-topology
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user 504602
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4
votes
Accepted
Embeddings of pseudo metric spaces into seminormed Spaces
Yes. If $(X, d)$ is a pseudometric space, then we may may define an equivalence relation $\sim$ on $X$ by $x \sim y$ iff $d(x, y) = 0$. Then $(X/{\sim}, d’)$ with $d’([x], [y]) = d(x, y)$ is a metric …
- 2,830
2
votes
A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
For the first question, yes, $S \setminus A$ is contractible. Rewrite the standard basis so that it is indexed by $\mathbb{Z}$. Let $U$ be the bilateral shift. Then $U$ is an isomorphism on $\ell^p$ f …
- 2,830
4
votes
Accepted
Approximating continuous functions from $K\times L$ into $[0,1]$
The following provides a construction of $g_i, h_i$ satisfying the required conditions (the ranges of these functions can even be in $[0, 1]$):
Claim: For each $k \in K$, there exists an open neighbor …
- 2,830
1
vote
Accepted
Signed measures on algebras (fields) and their boundedness properties
I already answered on MSE. Here is the answer, copied here as well:
Here is a counterexample, if I did not misunderstand the notations:
Let $X = \mathbb{N} \cup \{\infty\}$ be the one-point compactifi …
- 2,830
4
votes
Accepted
Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
The space is hyperconnected and thus also connected. Indeed, it suffices to show that, for any $x_1, \cdots, x_n \in \mathcal{P}(\omega)/(\text{fin})$ which are not $[\omega]$, and for any $y_1, \cdot …
- 2,830