Does PCA preserve linear separability for every linearly separable set? After using PCA to reduce dimensionality, does PCA preserve linear separability for any linearly separable set? Will the data still be linearly separable after the transformation? I am thinking that it does preserve linear separability because the PCA just reduces the dimension and not the relationship between the points in terms of separability. Am I on the right line of thinking?

No, reducing dimensionality with PCA will only maximize variance, which may or may not translate to linear separability.
Here are two visualizations of variance and separability in opposition. In both cases, the discriminative information lies primarily along the low-variance axis, which would get discarded by rote dimensionality reduction.

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*Minimal example of three data points (adapted from whuber's comment)



*Example of parallel, elongated Gaussian clusters (adapted from Dikran's answer)

import numpy as np
from sklearn.decomposition import PCA

# generate gaussian samples (n by 2)
rng = np.random.default_rng(4)
n = 100
X = rng.standard_normal((n, 2)) @ rng.random((2, 2))

# peel classes apart (+1 and -1 vertically)
y = 2 * np.arange(X.shape[0]) // n
X[:, 1] = np.where(y, X[:, 1] - 1, X[:, 1] + 1)

# project X
mu = X.mean(axis=0)
X -= mu
pca = PCA(n_components=2).fit(X) # explained variance ratios: [0.83267962, 0.16732038]
X_pca = pca.transform(X)

# reduce dimensionality via PC1
X_pc1 = X_pca[:, [0]] @ pca.components_[[0], :] + mu

# reduce dimensionality via PC2
X_pc2 = X_pca[:, [1]] @ pca.components_[[1], :] + mu

